Statistics for Business and Economics: Random Variables:Continuous
|
|
- Augustus Carter
- 5 years ago
- Views:
Transcription
1 Statistics for Business and Economics: Random Variables:Continuous STT 315: Section 107 Acknowledgement: I d like to thank Dr. Ashoke Sinha for allowing me to use and edit the slides. Murray Bourne (interactive math) and digitalfirst.bfwpub.com for applets
2 Probability Distribution The histogram of the random variable The plot probability density or relative frequency (p(x)) against the value of random variable (x). 2
3 Probability distribution Let X be any random variable with real number values. a and b are fixed numbers. P( a X b) = Area under the histogram of Y between, including a and between, including b Case discrete: X takes discrete values like 0,1,2,3,4,.10. X ~ Bin(10,0.5) P(1 X 2) =Area of rectangle_1 + area of rectangle_2 =height of rectangle_1* width rectangle_1 + height of rectangle_2* width rectangle_2 3
4 = binompdf(10,0.5,1)*width of rectangle 1 + binompdf(10,0.5,2)*width of rectangle 2 = * * 1 = = Binomcdf(10,0.5,2) - Binomcdf(10,0.5,0) 4
5 Case Continuous: Let Y be a continuous random variable eg: Y follows normal distribution P( 1 X 2)= 2 dx. 1 f x = = Integral of f(x) i.e probability density function from 1 to 2. = e (x μ)2 2σ 2 σ 2π dx
6
7 Normal Distribution 7
8 Normal random variable A normal random variable X is a continuous random variable has a probability distribution which is bell-shaped, i.e., unimodal, symmetric. In many data-sets, the histogram is bell-shaped. These data-sets can be modeled using normal distribution. Example :Height of population, shoe size, intelligence, error in measurement 8
9 Normal distribution Normal distribution is identified by its mean (μ) and standard deviation (σ). The form of the normal curve [probability density function] is defined for all real x, i.e. < x <, f x = 1 (x μ) 2 σ 2π e 2σ 2, where π = and e = A normal random variable with μ = 0, σ = 1 is called a standard normal random variable. If X is normal with mean mean (μ) and standard deviation σ, then Z = X μ is standard normal. σ 9
10 Normal graph effect of mean and standard deviations Go to above applet for interactive demo
11 Computing normal probabilities Since normal random variable is continuous P X = x = 0 for all x. Thus for any two numbers a and b, P a < X < b = P a < X b = P a X < b = P a X b. We shall use TI 83/84 for computation. We generally face two type of problems: To compute P a < X < b [use normalcdf]; Given the value of p finding x such that p = P X x [use invnorm]. 11
12 Approximately what percent of U.S. women do you expect to be between 66 in and 67 in tall? Heights of adult women are normally distributed with mean of 63.6 in, standard deviation of 2.5 in. Use TI 83/84 Plus. Press [2nd] & [VARS] (i.e. [DISTR]) Select 2: normalcdf Format of command: normalcdf(lower bound, upper bound, mean, std.dev.) For this problem: normalcdf(66, 67, 63.6, 2.5) = i.e. about 8.2% of adult U.S. women have heights between 66 in and 67 in. 12
13 P(1 X 2) = P(X=1) + P(1<X<2) +P(X=2) = Area of red + Area of yellow + Area of green = 0 + P(1<Y<2) + 0 Area of red, green rectangle = Height *( width= 0) =0
14 Approximately what percent of U.S. women do you expect to be less than 64 in tall? Heights of adult women are normally distributed with mean of 63.6 in, standard deviation of 2.5 in. Note that here upper bound is 64, but there is no mention of lower bound. So it is infinity (negative) So take a very small value for lower bound of your choice, say -10^10 which represents infinity For this problem normalcdf(-10^10, 64, 63.6, 2.5) = i.e. about 56.4% of adult U.S. women have heights less than 64 in. 14
15 Approximately what percent of U.S. women do you expect to be more than 58 in tall? Heights of adult women are normally distributed with mean of 63.6 in, standard deviation of 2.5 in. Note that here lower bound is 58, but there is no mention of upper bound. So it is + infinity (positive) So take a very high value for upper bound of your choice, say 10^10 which represents + infinity For this problem normalcdf(58, 10^10, 63.6, 2.5) = i.e. 98.7% of adult U.S. women have heights more than 58 in. 15
16 What about men s height? Heights of adult men are normally distributed with mean of 69 in, standard deviation of 2.8 in. normalcdf(60, 10^10, 69, 2.8) = Hence 99.9% adult male will have height more than 60 in. normalcdf(64, 10^10, 69, 2.8) = So 96.3% adult male will have height more than 64 in. Thus for U.S. Army height restriction for women is more restrictive compared to men. But for U.S. Marine height restriction for men is more restrictive compared to women. 16
17 Below what height 80% of U.S. men do have their heights? Heights of adult men are normally distributed with mean of 69 in, standard deviation of 2.8 in. The question is to find the height x such that {Percent of men s height < x} = 80% = 0.8. Use TI 83/84 Plus. Press [2nd] & [VARS] (i.e. [DISTR]) Select 3: invnorm Format of command: invnorm(fraction, mean, std.dev.) For this problem: invnorm(0.8, 69, 2.8) = i.e. 80% of U.S. men have heights less than in. 17
18 Remark: invnorm invnorm only considers percentage or fraction in the lower tail of normal distribution. For example, suppose the question is Above what height 10% of U.S. men do have their heights? Notice here the question is find the height x such that {Percent of men s height > x} = 10% = 0.1. This means {Percent of men s height < x} = (100-10)% = 90% = 0.9. For this problem: invnorm(0.9, 69, 2.8) = i.e. 90% of U.S. men have heights less than in, i.e. 10% of U.S. men have heights more than in. 18
19 Normal approximation of binomial distribution Suppose X~Bin n, p. Hence μ = np, σ X = npq. If n is very large, then the probability distribution of X can be approximated by normal distribution with μ = np, σ X = npq. However, X being binomially distributed is a discrete random variable, whereas normal distribution is continuous. So we need a continuity correction. If n is very large, then we compute as follows: P Bin n, p r. v. x P N np, npq r. v. x
20 How large n should be? To apply normal approximation to binomial distribution n should be so large that the interval np ± 3 npq should lie in the range 0 and n. Eg: n=1000 p=0.7 then np ± 3 npq =700 ± 3*14.49 Here , lie between 0 and n=1000 The applet demo : 20
21 Example Let X be binomially distributed with n = 50, p = 0.2. Here μ = np = 10, σ = npq = Thus np ± 3 npq = 1.515, 18.49, which lies in the range 0 and n = 50. Hence n is large enough. P X 16 P N 10, = normalcdf 100, 16.5, 10, = P X < 16 = P(X 15) P N 10, = normalcdf 100, 15.5, 10, = P X = 16 = P X 16 P X < = P X > 12 = 1 P X 12 1 PሾN 10, ሿ = 1 normalcdf 100, 12.5, 10, = P X 12 = 1 P X < 12 = 1 P X 11 1 normalcdf 100, 11.5, 10, =
22 Example Suppose X is normally distributed with mean μ and standard deviation σ. What is the probability that the value of X will be within 1.5 standard deviation from the mean? i.e. P μ 1.5σ X μ + 1.5σ =? Solution: Remember that Z = X μ The z-score of μ 1.5σ is = the z-score of μ + 1.5σ is = σ μ 1.5σ μ σ μ+1.5σ μ σ is standard normal. = 1.5 and = 1.5. So, P μ 1.5σ X μ + 1.5σ = P( 1.5 Z 1.5 ) =normalcdf(-1.5,1.5,0,1)=
23 Sum of independent random variables 23
24 Combining Random Variables Let X and Y be two random variables. Then E(X ± Y) = E(X) ± E(Y). If further X and Y are independent, then V(X ± Y) = V(X) + V(Y). Notice that for variance both have a plus sign on the right hand side. For expectation, independence assumption is not necessary, but for the above variance formula it is required. Variance for dependent case will not be treated in this course. 24
25 Example Suppose X and Y are two independent random variables with E(X) = 4, V(X) = 2, E(Y) = -3, V(Y) = 4. Then E(X+Y) = E(X)+E(Y) = 4+(-3) = 1. V(X+Y) = V(X)+V(Y) = 2+4 = 6. σ(x+y) = std. dev. of (X+Y) = V(X+Y) = 6 = E(X-Y) = E(X)-E(Y) = 4-(-3) = 7. V(X-Y) = V(X)+V(Y) = 2+4 = 6. σ(x-y) = std. dev. of (X-Y) = V(X Y) = 6 =
26 Example Suppose X and Y are two independent random variables with Then E(3X-2Y) = E(3X) - E(2Y) = 3E(X) - 2E(Y) = (-3) = = 18. E(X) = 4, V(X) = 2, E(Y) = -3, V(Y) = 4. 26
27 Example Suppose X and Y are two independent random variables with E(X) = 4, V(X) = 2, E(Y) = -3, V(Y) = 4. Then V(3X-2Y) = V(3X) + V(2Y) = 3 2 V(X) V(Y) = = = 34. σ(3x-2y) = std. dev. of (3X-2Y) = V(3X-2Y) = 34 =
28 Example These formulas can be extended to more than two random variables. Suppose we have the following information about random variables X, Y and Z. X, Y and Z are independent, and Random variables Expectations E(X + Y - Z) = E(X) + E(Y) E(Z) = (-4) = -11. V(X + Y - Z) = V(X) + V(Y) + V(Z) = = 12. σ(x + Y - Z) = 12 = Variances X -4 2 Y 2 6 Z
29 Another Example Suppose X, Y and Z are independent, and Random variables Expectations Standard deviations X 0 1 Y Z Notice that here we are given the standard deviations (not the variances). E(Y Z X) = E(Y) E(Z) E(X) = = V(Y Z X) = V(X) + V(Y) + V(Z) = = 35. σ(y Z X) = 35 =
30 Sum of independent normal random variables Suppose X is normal with mean μ 1 and variance σ 1 2, Y is normal with mean μ 2 and variance σ 2 2, and X and Y are independent of each other. Then X + Y is normally distributed with mean μ 1 + μ 2 and variance σ σ 2 2. That means E X + Y = μ 1 + μ 2. var X + Y = σ σ 2 2. σ X + Y = σ σ
31 Example Suppose the monthly revenue in investment A is normally distributed with mean $25 and std.dev. $8, and that in investment B is normally distributed with mean $31 and std.dev. $10. If you have both investments, what is the probability that your total monthly revenue will be more than $75? The total monthly revenue will be normally distributed with mean $(25+31)=$56, and std. dev = $ So probability that your total monthly revenue will be more than $75 is = normalcdf(75,1000,56,12.806) =
32 Example Suppose the monthly revenue in investment A is normally distributed with mean $25 and std.dev. $8, and that in investment B is normally distributed with mean $31 and std.dev. $10. Above what value 80% of total monthly revenue will lie? The total monthly revenue will be normally distributed with mean $(25+31)=$56, and std. dev = $ If x is the value above which 80% of total monthly revenue will lie, then 20% of total monthly revenue will lie below x. Thus x = invnorm(0.2,56,12.806) = $
33 Uniform distribution 33
34 Uniform distribution A continuous random variable X is uniformly distributed in the interval ሾa, bሿ if its probability density function is f x = 1 b a, for a x b. In this case, if a l < u b o P l < X < u = u l b a, o E X = a+b 2, o σ X = b a
35 Example If price of gas (X) in East Lansing has uniform distribution in the interval $[3.45, 3.95] per gallon. Probability that gas price will be between $3.50 and $3.60 = P 3.50 X 3.60 = = 0.2. Probability that gas price will be less than $3.70 = P( X < 3.70 ) = P 3.45 X < 3.70 = = Probability that gas price will be more than $3.90 = P( X > 3.90 ) = P 3.90 < X 3.95 = = Probability that gas price will be $3.82 = P X = 3.82 = 0, because X is a continuous random variable. 35
36 Example If price of gas (X) in East Lansing has uniform distribution in the interval $[3.45, 3.95] per gallon. The expected gas price in East Lansing is = $3.70. = The standard deviation of gas price in East Lansing is = $ =
37 Sampling distributions 37
38 Remember Population is the complete set of all items that we are interested in studying. Parameters are the values we calculate from the population data. Population mean (for quantitative variables), population proportion (categorical variables) etc. are the examples of parameters. A sample is a subset of the population. Statistics are values we compute from sample data. Sample mean, sample proportion etc. are the examples of statistics. Our goal is to make inference on parameters based on relevant statistics. 38
39 An example Consider a population with 10 individuals with the following smoking habit: Individual #: Smoking habit: N N N N S S N N S N where S = smoker, and N = non-smoker. So 3 out of 10 people in the population is smoker. Here the population proportion of smoker is: p
40 An example Suppose we decide to estimate population proportion on the basis of a sample proportion. Suppose simple random samples of size 4 (with replacement) are considered. Individuals selected Smoking habit Sample proportion (2, 4, 4, 9) (N, N, N, S) 1/4 = 0.25 (4, 7, 8, 10) (N, N, N, N) 0/4 = 0 (5, 6, 8, 8) (S, S, N, N) 2/4 = 0.5 Notice that the sample proportion s value depends on the sample selected, but the population proportion s value is fixed. 40
41 Few questions Can we justify the use of sample proportion as an estimator of population proportion? What can we expect about the value of sample proportion when population proportion (p) is 0.3? Does this behavior depend on the value of p? What is the margin of error, if we estimate p with sample proportion? (To be answered in a later lecture.) As sample proportion is a variable, what is its distribution? 41
42 Few questions Does it matter how the sample is selected? Does the sample size matter? Is this a problem of population proportion only? Or do we face it for other parameters also? This is a problem for all parameters, which are fixed in value for a particular population. The value of any statistic changes with the sample selected. 42
43 Sampling Distribution As any statistic s value changes with the selected sample, so statistic is a itself a random variable. The probability distribution of a sample statistic is called the sampling distribution of the statistic. In this course we shall study sampling distributions of sample proportion and sample mean. 43
44 Sampling method and sample size Samples must be independent. Simple random sampling with replacement ensures independence. Holds (approximately) also for without replacement sampling as long as the sample size is smaller than 10% of the population size. Sample size must be large enough. What is large enough depends on the statistic we are considering, i.e. different rules of large enough for sample proportion and sample mean. It is the sample size what is important, NOT what fraction of population is sampled. 44
45 Sampling distribution of sample proportion ( p) 45
46 Sampling distribution of sample proportion Consider in a population a categorical variable with two categories: success and failure. e.g., smoking habit variable the level smoker can be considered as success, and non-smoker as failure. Let p be the population proportion of success. A random sample from the population is drawn. Observations in the sample are independent. Sample size is n. Let x be the number of success in the sample. Then sample proportion of success is p Ƹ = x n. 46
47 Sampling distribution of sample proportion The expected value of pƹ is equal to p, i.e. E The standard deviation of pƹ is σ p = p(1 p) p Ƹ = p. If we repeatedly simulate the selection of samples from the population with large enough sample size, the distribution of the sample proportions we found in the samples will be roughly normally distributed and the distribution will be N p, n p(1 p) n.. 47
48 Sampling distribution of sample proportion When is sample large enough for the last result to hold? If n is so large that np > 9, and n(1 p) > 9. This is covered if the number of successes and failures are both at least
49 Example One Of all the cars on the highway, about 80% exceed the speed limit. If we clock the next 50 cars that pass, what might we expect to find? Is the independence condition met? Most likely NO, because the cars moving at the same time may influence each others behavior. Suppose we randomly select 50 cars that pass. Is the independence condition met? Yes. 49
50 Example One Of all the cars on the highway, about 80% exceed the speed limit. Suppose we randomly select 50 cars that pass. Is sample size large enough condition met? a) Yes b) No Because, np = = 40 > 9, and n(1-p) = = 10 > 9. 50
51 Example One Of all the cars on the highway, about 80% exceed the speed limit. Suppose we randomly select 50 cars that pass. What is the expected proportion of cars in the sample to exceed the speed limit? A. 20% B. 80% C. 2.83% D % 51
52 Example One Of all the cars on the highway, about 80% exceed the speed limit. Suppose we randomly select 50 cars that pass. What is the standard deviation of the sample proportion of cars exceeding the speed limit? A. 20 B. 80 C D
53 Example One Of all the cars on the highway, about 80% exceed the speed limit. Suppose we randomly select 50 cars that pass. What is the chance that more than 90% of cars in the sample exceeded the speed limit? A B C D E normalcdf(0.9,100,0.8,0.057) =
54 Sampling distribution of sample mean (ഥx) 54
55 Sampling distribution of sample mean Suppose the mean of population distribution is µ and standard deviation σ. A random sample from the population is drawn. Observations in the sample are independent. Sample size is n. Let the sample mean be xҧ The expected value of xҧ is equal to µ. The standard deviation of xҧ is σ. n 55
56 Central Limit Theorem (CLT) If we repeatedly simulate the selection of samples from the population with large enough sample size, the distribution of the sample mean in random sampling roughly follows a normal model and the distribution will be N(μ, σ n ). The larger the sample size, the closer to normal the distribution will be. But how large is large enough to apply CLT? We can use CLT if n
57 Example Two At birth, babies average 7.8 pounds, with a standard deviation of 2.1 pounds. A random sample of 34 babies born to mothers living near a factory that might be polluting the air and water shows a mean birth-weight of only 7.2 pounds. What is the expected value of sample mean? A. 34 lb B. 7.2 lb C. 7.8 lb D. 2.1 lb 57
58 Example Two At birth, babies average 7.8 pounds, with a standard deviation of 2.1 pounds. A random sample of 34 babies born to mothers living near a factory that might be polluting the air and water shows a mean birth-weight of only 7.2 pounds. What is the standard deviation of sample mean? A lb B. 7.2 lb C lb D. 2.1 lb 58
59 Example Two At birth, babies average 7.8 pounds, with a standard deviation of 2.1 pounds. A random sample of 34 babies born to mothers living near a factory that might be polluting the air and water shows a mean birth-weight of only 7.2 pounds. What is the chance that the sample mean is lower than 7.2 lbs? A B C D normalcdf(-100, 7.2, 7.8, 0.36) =
60 Example If price of gas (X) in East Lansing has uniform distribution in the interval $[3.45, 3.95] per gallon. Remember that μ = E X = $3.70, σ X = $ Suppose we collect gas prices from 35 gas stations of East Lansing. The expected sample average of gas prices is E തX = μ = $3.70. The standard deviation of sample average of gas prices is σ തX = σ n = = $ Since n > 30, we have തX~N 3.70, The chance that the sample average will be less than $3.65 is P തX < 3.65 = normalcdf 100, 3.65, 3.70, =
As you draw random samples of size n, as n increases, the sample means tend to be normally distributed.
The Central Limit Theorem The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of statistics. The clt says that if we collect samples of size n with a "large enough
More informationCH 5 Normal Probability Distributions Properties of the Normal Distribution
Properties of the Normal Distribution Example A friend that is always late. Let X represent the amount of minutes that pass from the moment you are suppose to meet your friend until the moment your friend
More informationWeek 7. Texas A& M University. Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4
Week 7 Oğuz Gezmiş Texas A& M University Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 1 / 19
More informationBusiness Statistics 41000: Probability 4
Business Statistics 41000: Probability 4 Drew D. Creal University of Chicago, Booth School of Business February 14 and 15, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
More informationChapter 7 1. Random Variables
Chapter 7 1 Random Variables random variable numerical variable whose value depends on the outcome of a chance experiment - discrete if its possible values are isolated points on a number line - continuous
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationChapter 7 Sampling Distributions and Point Estimation of Parameters
Chapter 7 Sampling Distributions and Point Estimation of Parameters Part 1: Sampling Distributions, the Central Limit Theorem, Point Estimation & Estimators Sections 7-1 to 7-2 1 / 25 Statistical Inferences
More informationStatistics (This summary is for chapters 17, 28, 29 and section G of chapter 19)
Statistics (This summary is for chapters 17, 28, 29 and section G of chapter 19) Mean, Median, Mode Mode: most common value Median: middle value (when the values are in order) Mean = total how many = x
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 7 The Normal Distribution Part 1 Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education
More informationStatistics (This summary is for chapters 18, 29 and section H of chapter 19)
Statistics (This summary is for chapters 18, 29 and section H of chapter 19) Mean, Median, Mode Mode: most common value Median: middle value (when the values are in order) Mean = total how many = x n =
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 6 Normal Probability Distributions 6-1 Overview 6-2 The Standard Normal Distribution
More informationHomework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a
Homework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a Announcements: There are some office hour changes for Nov 5, 8, 9 on website Week 5 quiz begins after class today and ends at
More information7 THE CENTRAL LIMIT THEOREM
CHAPTER 7 THE CENTRAL LIMIT THEOREM 373 7 THE CENTRAL LIMIT THEOREM Figure 7.1 If you want to figure out the distribution of the change people carry in their pockets, using the central limit theorem and
More informationLecture 6: Chapter 6
Lecture 6: Chapter 6 C C Moxley UAB Mathematics 3 October 16 6.1 Continuous Probability Distributions Last week, we discussed the binomial probability distribution, which was discrete. 6.1 Continuous Probability
More informationHomework: Due Wed, Feb 20 th. Chapter 8, # 60a + 62a (count together as 1), 74, 82
Announcements: Week 5 quiz begins at 4pm today and ends at 3pm on Wed If you take more than 20 minutes to complete your quiz, you will only receive partial credit. (It doesn t cut you off.) Today: Sections
More informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 6 The Normal Distribution 2 Objectives Identify distributions as symmetrical or skewed. Identify the properties of the normal distribution. Find
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
More informationContinuous Probability Distributions & Normal Distribution
Mathematical Methods Units 3/4 Student Learning Plan Continuous Probability Distributions & Normal Distribution 7 lessons Notes: Students need practice in recognising whether a problem involves a discrete
More informationData Analysis and Statistical Methods Statistics 651
Review of previous lecture: Why confidence intervals? Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Suppose you want to know the
More informationMATH 104 CHAPTER 5 page 1 NORMAL DISTRIBUTION
MATH 104 CHAPTER 5 page 1 NORMAL DISTRIBUTION We have examined discrete random variables, those random variables for which we can list the possible values. We will now look at continuous random variables.
More informationUsing the Central Limit Theorem It is important for you to understand when to use the CLT. If you are being asked to find the probability of the
Using the Central Limit Theorem It is important for you to understand when to use the CLT. If you are being asked to find the probability of the mean, use the CLT for the mean. If you are being asked to
More informationVersion A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
More informationSection 7.5 The Normal Distribution. Section 7.6 Application of the Normal Distribution
Section 7.6 Application of the Normal Distribution A random variable that may take on infinitely many values is called a continuous random variable. A continuous probability distribution is defined by
More informationNormal Probability Distributions
C H A P T E R Normal Probability Distributions 5 Section 5.2 Example 3 (pg. 248) Normal Probabilities Assume triglyceride levels of the population of the United States are normally distributed with a mean
More informationContinuous Random Variables and the Normal Distribution
Chapter 6 Continuous Random Variables and the Normal Distribution Continuous random variables are used to approximate probabilities where there are many possible outcomes or an infinite number of possible
More informationOverview. Definitions. Definitions. Graphs. Chapter 5 Probability Distributions. probability distributions
Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 5-5 The Poisson Distribution
More informationBusiness Statistics 41000: Probability 3
Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404
More informationChapter 8 Estimation
Chapter 8 Estimation There are two important forms of statistical inference: estimation (Confidence Intervals) Hypothesis Testing Statistical Inference drawing conclusions about populations based on samples
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 207 Homework 5 Drew Armstrong Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Section 3., Exercises 3, 0. Section 3.3, Exercises 2, 3, 0,.
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 4 Random Variables & Probability Distributions Content 1. Two Types of Random Variables 2. Probability Distributions for Discrete Random Variables 3. The Binomial
More informationExamples of continuous probability distributions: The normal and standard normal
Examples of continuous probability distributions: The normal and standard normal The Normal Distribution f(x) Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread.
More informationClass 16. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 16 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 013 by D.B. Rowe 1 Agenda: Recap Chapter 7. - 7.3 Lecture Chapter 8.1-8. Review Chapter 6. Problem Solving
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationPart V - Chance Variability
Part V - Chance Variability Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Part V - Chance Variability 1 / 78 Law of Averages In Chapter 13 we discussed the Kerrich coin-tossing experiment.
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 27 Continuous
More informationChapter 6. The Normal Probability Distributions
Chapter 6 The Normal Probability Distributions 1 Chapter 6 Overview Introduction 6-1 Normal Probability Distributions 6-2 The Standard Normal Distribution 6-3 Applications of the Normal Distribution 6-5
More informationShifting and rescaling data distributions
Shifting and rescaling data distributions It is useful to consider the effect of systematic alterations of all the values in a data set. The simplest such systematic effect is a shift by a fixed constant.
More informationCentral Limit Theorem, Joint Distributions Spring 2018
Central Limit Theorem, Joint Distributions 18.5 Spring 218.5.4.3.2.1-4 -3-2 -1 1 2 3 4 Exam next Wednesday Exam 1 on Wednesday March 7, regular room and time. Designed for 1 hour. You will have the full
More informationBinomial and Normal Distributions. Example: Determine whether the following experiments are binomial experiments. Explain.
Binomial and Normal Distributions Objective 1: Determining if an Experiment is a Binomial Experiment For an experiment to be considered a binomial experiment, four things must hold: 1. The experiment is
More informationCentral Limit Theorem
Central Limit Theorem Lots of Samples 1 Homework Read Sec 6-5. Discussion Question pg 329 Do Ex 6-5 8-15 2 Objective Use the Central Limit Theorem to solve problems involving sample means 3 Sample Means
More informationCHAPTER 5 SAMPLING DISTRIBUTIONS
CHAPTER 5 SAMPLING DISTRIBUTIONS Sampling Variability. We will visualize our data as a random sample from the population with unknown parameter μ. Our sample mean Ȳ is intended to estimate population mean
More informationUsing the Central Limit
Using the Central Limit Theorem By: OpenStaxCollege It is important for you to understand when to use the central limit theorem. If you are being asked to find the probability of the mean, use the clt
More informationLecture 12. Some Useful Continuous Distributions. The most important continuous probability distribution in entire field of statistics.
ENM 207 Lecture 12 Some Useful Continuous Distributions Normal Distribution The most important continuous probability distribution in entire field of statistics. Its graph, called the normal curve, is
More informationCHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS A random variable is the description of the outcome of an experiment in words. The verbal description of a random variable tells you how to find or calculate
More informationThe Normal Probability Distribution
102 The Normal Probability Distribution C H A P T E R 7 Section 7.2 4Example 1 (pg. 71) Finding Area Under a Normal Curve In this exercise, we will calculate the area to the left of 5 inches using a normal
More informationLecture 9. Probability Distributions. Outline. Outline
Outline Lecture 9 Probability Distributions 6-1 Introduction 6- Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7- Properties of the Normal Distribution
More informationStatistics for Business and Economics: Random Variables (1)
Statistics for Business and Economics: Random Variables (1) STT 315: Section 201 Instructor: Abdhi Sarkar Acknowledgement: I d like to thank Dr. Ashoke Sinha for allowing me to use and edit the slides.
More informationChapter 6 Continuous Probability Distributions. Learning objectives
Chapter 6 Continuous s Slide 1 Learning objectives 1. Understand continuous probability distributions 2. Understand Uniform distribution 3. Understand Normal distribution 3.1. Understand Standard normal
More informationTheoretical Foundations
Theoretical Foundations Probabilities Monia Ranalli monia.ranalli@uniroma2.it Ranalli M. Theoretical Foundations - Probabilities 1 / 27 Objectives understand the probability basics quantify random phenomena
More informationAMS7: WEEK 4. CLASS 3
AMS7: WEEK 4. CLASS 3 Sampling distributions and estimators. Central Limit Theorem Normal Approximation to the Binomial Distribution Friday April 24th, 2015 Sampling distributions and estimators REMEMBER:
More informationNORMAL RANDOM VARIABLES (Normal or gaussian distribution)
NORMAL RANDOM VARIABLES (Normal or gaussian distribution) Many variables, as pregnancy lengths, foot sizes etc.. exhibit a normal distribution. The shape of the distribution is a symmetric bell shape.
More information8.1 Binomial Distributions
8.1 Binomial Distributions The Binomial Setting The 4 Conditions of a Binomial Setting: 1.Each observation falls into 1 of 2 categories ( success or fail ) 2 2.There is a fixed # n of observations. 3.All
More information6.1 Discrete & Continuous Random Variables. Nov 4 6:53 PM. Objectives
6.1 Discrete & Continuous Random Variables examples vocab Objectives Today we will... - Compute probabilities using the probability distribution of a discrete random variable. - Calculate and interpret
More informationSTATISTICAL DISTRIBUTIONS AND THE CALCULATOR
STATISTICAL DISTRIBUTIONS AND THE CALCULATOR 1. Basic data sets a. Measures of Center - Mean ( ): average of all values. Characteristic: non-resistant is affected by skew and outliers. - Median: Either
More informationLecture 9. Probability Distributions
Lecture 9 Probability Distributions Outline 6-1 Introduction 6-2 Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7-2 Properties of the Normal Distribution
More informationStatistical Methods in Practice STAT/MATH 3379
Statistical Methods in Practice STAT/MATH 3379 Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Overview 6.1 Discrete
More informationOverview. Definitions. Definitions. Graphs. Chapter 4 Probability Distributions. probability distributions
Chapter 4 Probability Distributions 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions 4-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 4-5 The Poisson Distribution
More informationUnit2: Probabilityanddistributions. 3. Normal distribution
Announcements Unit: Probabilityanddistributions 3 Normal distribution Sta 101 - Spring 015 Duke University, Department of Statistical Science February, 015 Peer evaluation 1 by Friday 11:59pm Office hours:
More informationThe Normal Distribution
5.1 Introduction to Normal Distributions and the Standard Normal Distribution Section Learning objectives: 1. How to interpret graphs of normal probability distributions 2. How to find areas under the
More informationNormal distribution. We say that a random variable X follows the normal distribution if the probability density function of X is given by
Normal distribution The normal distribution is the most important distribution. It describes well the distribution of random variables that arise in practice, such as the heights or weights of people,
More informationMAS1403. Quantitative Methods for Business Management. Semester 1, Module leader: Dr. David Walshaw
MAS1403 Quantitative Methods for Business Management Semester 1, 2018 2019 Module leader: Dr. David Walshaw Additional lecturers: Dr. James Waldren and Dr. Stuart Hall Announcements: Written assignment
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
More informationChapter 4 Probability Distributions
Slide 1 Chapter 4 Probability Distributions Slide 2 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions 4-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 4-5
More informationMath 120 Introduction to Statistics Mr. Toner s Lecture Notes. Standardizing normal distributions The Standard Normal Curve
6.1 6.2 The Standard Normal Curve Standardizing normal distributions The "bell-shaped" curve, or normal curve, is a probability distribution that describes many reallife situations. Basic Properties 1.
More informationcontinuous rv Note for a legitimate pdf, we have f (x) 0 and f (x)dx = 1. For a continuous rv, P(X = c) = c f (x)dx = 0, hence
continuous rv Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f(x) such that for any two numbers a and b with a b, P(a X b) = b a f (x)dx.
More informationUsing the Central Limit Theorem
OpenStax-CNX module: m46992 1 Using the Central Limit Theorem OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 It is important for
More information5.4 Normal Approximation of the Binomial Distribution
5.4 Normal Approximation of the Binomial Distribution Bernoulli Trials have 3 properties: 1. Only two outcomes - PASS or FAIL 2. n identical trials Review from yesterday. 3. Trials are independent - probability
More information15.063: Communicating with Data Summer Recitation 4 Probability III
15.063: Communicating with Data Summer 2003 Recitation 4 Probability III Today s Content Normal RV Central Limit Theorem (CLT) Statistical Sampling 15.063, Summer '03 2 Normal Distribution Any normal RV
More informationIntroduction to Business Statistics QM 120 Chapter 6
DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Chapter 6: Continuous Probability Distribution 2 When a RV x is discrete, we can
More informationLecture 8. The Binomial Distribution. Binomial Distribution. Binomial Distribution. Probability Distributions: Normal and Binomial
Lecture 8 The Binomial Distribution Probability Distributions: Normal and Binomial 1 2 Binomial Distribution >A binomial experiment possesses the following properties. The experiment consists of a fixed
More informationMath Week in Review #10. Experiments with two outcomes ( success and failure ) are called Bernoulli or binomial trials.
Math 141 Spring 2006 c Heather Ramsey Page 1 Section 8.4 - Binomial Distribution Math 141 - Week in Review #10 Experiments with two outcomes ( success and failure ) are called Bernoulli or binomial trials.
More informationBasic Data Analysis. Stephen Turnbull Business Administration and Public Policy Lecture 4: May 2, Abstract
Basic Data Analysis Stephen Turnbull Business Administration and Public Policy Lecture 4: May 2, 2013 Abstract Introduct the normal distribution. Introduce basic notions of uncertainty, probability, events,
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 28 One more
More informationSection Introduction to Normal Distributions
Section 6.1-6.2 Introduction to Normal Distributions 2012 Pearson Education, Inc. All rights reserved. 1 of 105 Section 6.1-6.2 Objectives Interpret graphs of normal probability distributions Find areas
More informationData that can be any numerical value are called continuous. These are usually things that are measured, such as height, length, time, speed, etc.
Chapter 8 Measures of Center Data that can be any numerical value are called continuous. These are usually things that are measured, such as height, length, time, speed, etc. Data that can only be integer
More informationME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
More informationStatistics for IT Managers
Statistics for IT Managers 95-796, Fall 212 Course Overview Instructor: Daniel B. Neill (neill@cs.cmu.edu) TAs: Eli (Han) Liu, Kats Sasanuma, Sriram Somanchi, Skyler Speakman, Quan Wang, Yiye Zhang (see
More information8.1 Estimation of the Mean and Proportion
8.1 Estimation of the Mean and Proportion Statistical inference enables us to make judgments about a population on the basis of sample information. The mean, standard deviation, and proportions of a population
More informationChapter 5. Sampling Distributions
Lecture notes, Lang Wu, UBC 1 Chapter 5. Sampling Distributions 5.1. Introduction In statistical inference, we attempt to estimate an unknown population characteristic, such as the population mean, µ,
More informationStatistics, Their Distributions, and the Central Limit Theorem
Statistics, Their Distributions, and the Central Limit Theorem MATH 3342 Sections 5.3 and 5.4 Sample Means Suppose you sample from a popula0on 10 0mes. You record the following sample means: 10.1 9.5 9.6
More informationECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10
ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10 Fall 2011 Lecture 8 Part 2 (Fall 2011) Probability Distributions Lecture 8 Part 2 1 / 23 Normal Density Function f
More informationStat 101 Exam 1 - Embers Important Formulas and Concepts 1
1 Chapter 1 1.1 Definitions Stat 101 Exam 1 - Embers Important Formulas and Concepts 1 1. Data Any collection of numbers, characters, images, or other items that provide information about something. 2.
More information11.5: Normal Distributions
11.5: Normal Distributions 11.5.1 Up to now, we ve dealt with discrete random variables, variables that take on only a finite (or countably infinite we didn t do these) number of values. A continuous random
More informationPoint Estimation. Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage
6 Point Estimation Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Point Estimation Statistical inference: directed toward conclusions about one or more parameters. We will use the generic
More informationProbability Distributions II
Probability Distributions II Summer 2017 Summer Institutes 63 Multinomial Distribution - Motivation Suppose we modified assumption (1) of the binomial distribution to allow for more than two outcomes.
More informationNo, because np = 100(0.02) = 2. The value of np must be greater than or equal to 5 to use the normal approximation.
1) If n 100 and p 0.02 in a binomial experiment, does this satisfy the rule for a normal approximation? Why or why not? No, because np 100(0.02) 2. The value of np must be greater than or equal to 5 to
More informationPROBABILITY DISTRIBUTIONS. Chapter 6
PROBABILITY DISTRIBUTIONS Chapter 6 6.1 Summarize Possible Outcomes and their Probabilities Random Variable Random variable is numerical outcome of random phenomenon www.physics.umd.edu 3 Random Variable
More informationBinomial Distribution. Normal Approximation to the Binomial
Binomial Distribution Normal Approximation to the Binomial /29 Homework Read Sec 6-6. Discussion Question pg 337 Do Ex 6-6 -4 2 /29 Objectives Objective: Use the normal approximation to calculate 3 /29
More informationChapter 5 Normal Probability Distributions
Chapter 5 Normal Probability Distributions Section 5-1 Introduction to Normal Distributions and the Standard Normal Distribution A The normal distribution is the most important of the continuous probability
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter7 Probability Distributions and Statistics Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number of boys in
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability
More informationThe "bell-shaped" curve, or normal curve, is a probability distribution that describes many real-life situations.
6.1 6.2 The Standard Normal Curve The "bell-shaped" curve, or normal curve, is a probability distribution that describes many real-life situations. Basic Properties 1. The total area under the curve is.
More informationExpected Value of a Random Variable
Knowledge Article: Probability and Statistics Expected Value of a Random Variable Expected Value of a Discrete Random Variable You're familiar with a simple mean, or average, of a set. The mean value of
More informationLecture 5 - Continuous Distributions
Lecture 5 - Continuous Distributions Statistics 102 Colin Rundel January 30, 2013 Announcements Announcements HW1 and Lab 1 have been graded and your scores are posted in Gradebook on Sakai (it is good
More informationChapter 6: Normal Probability Distributions
Chapter 6: Normal Probability Distributions Section Title Notes Pages 1 Review & Preview 1 2 The Standard Normal Distribution 5 9 3 Applications of Normal Distributions 10 15 4 Sampling Distributions &
More informationThe graph of a normal curve is symmetric with respect to the line x = µ, and has points of
Stat 400, section 4.3 Normal Random Variables notes prepared by Tim Pilachowski Another often-useful probability density function is the normal density function, which graphs as the familiar bell-shaped
More informationProbability. An intro for calculus students P= Figure 1: A normal integral
Probability An intro for calculus students.8.6.4.2 P=.87 2 3 4 Figure : A normal integral Suppose we flip a coin 2 times; what is the probability that we get more than 2 heads? Suppose we roll a six-sided
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number
More information