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
Section 3.2-Measures of Central Tendency Definition The average or mean of the n numbers x 1, x 2,..., x n denoted by µ, is given by µ = x 1 + x 2 + + x n n In an examination given to a class of 6 students, the following test scores were obtained. 35 45 50 50 55 60. Find the average (or mean) score. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 2 / 19
Expected Value (Mean) Definition Let X denote the random variable that has values x 1, x 2,..., x n and let the associated probabilities be p 1, p 2,..., p n. The expected value or mean of the random variable X, denoted by E(X ), is E(X ) = x 1 p 1 + x 2 p 2 + + x n p n. Find the expected value of the random variable X having the probability distribution given in the following table. X -30-10 0 5 10 20 P(X) 0.15 0.25 0.05 0.10 0.25 0.20 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 3 / 19
In this bet select any of the 37 numbers. Making a 1 dollar bet results in winning back 36 dollars if your number occurs and, of course, you lose your 1 dollar if your number does not occur. Find the expected return. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 4 / 19
Expected Value of Binomial Distribution Definition The expected value of the binomial distribution with n trials and probability of success p in a single trial and q = 1 p is E(X ) = np. Find the expected value of the binomial distribution where n = 6 and p = 0.75. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 5 / 19
Median and Mode Definition 1) The median of a set of numerical data is the middle number when the numbers are arranged in order of size and there is an odd number of entries in the set. 2) In the case that the number of entries in the set is even, the median is the mean of the two middle numbers. 3) The mode of a set of observations is the observation that occurs more frequently than the others. 4) If the occurrence of two observations is the same and also greater than the frequency of occurrence of all the other observations, then we say the set is bimodal and has two modes. 5) If no one or two observations occurs more frequently than the others, we say that the set has no mode. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 6 / 19
Find the mean, median and the mode of the set of data {1, 2, 6, 1, 3, 6, 6, 1}. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 7 / 19
Find the mean, median and the mode of the set of data {12, 20, 310, 20, 18, 29, 20}. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 8 / 19
Variance and Standard Deviation Definition Let X denote the random variable that takes on the values x 1, x 2,..., x n and let the associated probabilities be p 1, p 2,..., p n. Then if µ = E(X ), the variance of the random variable X, denoted by Var(X ), is Var(X ) = (x 1 µ) 2 p 1 + (x 2 µ) 2 p 2 + + (x n µ) 2 p n. The standard deviation, denoted by σ(x ) is σ(x ) = Var(X ). Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 9 / 19
Remark Alternate Form of Variance: An alternate form of the variance is given by Var(X ) = (x1 2 p 1 + + xnp 2 n ) µ. Find the expected value (mean), variance and standard deviation of the random variable having the probability distribution given in the following table. r.v., x -2-1 1 2 P(X=x) 0.3 0.2 0.25 0.25 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 10 / 19
Variance of the Binomial Distribution Definition The variance of the binomial distribution with n trials and the probability of success equal to p and of failure equal to q is and the standard deviation is Var(X ) = npq. σ(x ) = npq. A new drug has been found to be effective in treating 75 % of the people afflicted by a certain disease. If the drug is administered to 800 people who have this disease, what are the mean(expected value) and the standard deviation of the number of people for whom the drug can be expected to be effective? Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 11 / 19
Remark Suppose you want to calculate the mean(expected value), variance and the standard deviation of the following table r.v., X 4 3 2 1 0 Frequency 25 11 3 1 0 For this first press STAT and select 1:Edit and press ENTER. Then type the values of X in the list to L1 and type the values of frequencies to L2. Press ENTER after you type each element in the list. Then press STAT again and right arrow. Go to CALC menu and select 1:1-Var Stats and then press enter and then write L1 by pressing 2nd first then 1 and then comma and L2 by pressing 2nd first then 2. Then press ENTER. The mean is given in the first line and the standard deviation is given in the 4th line. To find the variance press VARS button and choose 5: Statistics. Then choose 4:σx which will return to you the home screen and then press x 2 and ENTER to find variance. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 12 / 19
Section 3.4-The Normal Distribution In the previous sections, we focused especially on discrete random variables and represented them in a histogram. In this section, our aim is to understand continuous random variables. NOTE: The random variable associated with the normal distribution is designated by Z. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 13 / 19
Definition The random variable X has a normal distribution on the interval (, ) if the probability P(a X b) that X is between a and b is the area under the standard normal curve given by y = 1 σ (x µ) 2π exp 0.5[ σ ]2 on the interval [a, b], where π 3.14159 and exp 2.71828 and σ is the standard deviation and µ is the mean (expected value). Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 14 / 19
Remark Characteristic of the Normal Curve The graph of the normal curve has the following characteristics. 1) It is bell shaped. 2) It is symmetric about x = µ. 3) It lies above the x-axis. 4) It approaches but is never equal to 0 on both the positive and negative x-axis. 5) The area under the entire curve is exactly 1. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 15 / 19
Remark TI 83/84 calculators have three functions that are used for normal probability calculations. Access the Distributions menu by pressing 2nd and then VARS. The 1:normalpdf( command will find the normal probability density function for a given value of Z. The 2:normalcdf( command finds area under the normal probability density function. The first value entered is left endpoint and the second value entered is the right endpoint. If, for example, a normal probability distribution has a mean of 100 and a standard deviation of 15, we find P(70 X 80) as normalcdf(70,80,100,15). In general, it is given as normalcdf(leftpoint,rightpoint,mean,stand.dev). Suppose you want to find c such that P(X c) = 0.75 on a normal probability distribution with a mean µ of 20 and standard deviation σ of 5. Then again go to Distributions menu select 3:invNorm(. Enter invnorm(0.75,20,5) to find c. In general, it is given as invnorm(probability,mean,stand.dev). Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 16 / 19
Definition When σ = 1 and µ = 0, we call the distribution as standard normal distribution. Let Z be a random variable with standard normal distribution (µ = 0 and σ = 1) a) Find P(0.24 < Z < 1.48). b) Find P(Z < 1.24). c) Find P(Z > 0.5). d) Find c when P(Z c) = 0.72. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 17 / 19
Let X be a random variable. Find the following probabilities when µ and σ given. a) Find P(X 50) when µ = 38, σ = 8. b) Find c when P(Z c) = 0.97 when µ = 100, σ = 50. c) Find P(X 0.01) when µ = 0.006, σ = 0.002. d) Find P(10 X 20) when µ = 5, σ = 10. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 18 / 19
(a) Let Z be the standard normal variable. Find a such that P( a < Z < a) = 0.4131. (b) Let Z be the normal variable with mean µ = 60 and standard deviation σ = 3.5. Find A and B such that P(A < Z < B) = 0.7923 if A and B are symmetric about the mean. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 19 / 19