Section Random Variables and Histograms

Transcription

1 Section Random Variables and Histograms 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 define the random variable X to represent the number of times we get tails. Example 2: Suppose we roll a die until a 5 is facing up. Then we could define the random variable Y to represent the number of times we rolled the die. Example 3: Suppose a flashlight is left on until the battery runs out. Then we could define the random variable Z to represent the amount of time that passed. Types of Random Variables 1. A finite discrete random variable is one which can only take on a limited number of values that can be listed. 2. An infinite discrete random variable is one which can take on an unlimited number of values that can be listed in some sort of sequence. 3. A continuous random variable is one which takes on any of the infinite number of values in some interval of real numbers. (i.e. usually measurements) Example 4: Classify each of the following random variables as finite discrete, infinite discrete, or continuous and list/describe the possible values for the random variable. a) A drawer contains 15 red pens, 8 blue pens, and 5 black pens. An experiment consists of drawing pens out of the drawer with replacement until a red pen is drawn. Let X represent the number of pens drawn. b) An experiment consists of randomly selecting 3 pens without replacement from a conveyor belt. Let Y represent the total weight (in ounces) of the selected pens. c) An experiment consists of randomly selecting 3 pens with replacement from a drawer that contains 15 red pens, 8 blue pens, and 5 black pens. Let Z represent the number of red pens drawn. 1

2 Example 5: Referring to Example 1, find the probability distribution of the random variable X. Example 6: An experiment consists of randomly selecting a sample of 3 grapes out of a bowl that contains 20, of which 8 are rotten. Let X represent the number of rotten grapes in the sample. Find the probability distribution of X. Example 7: Blue Baker prides itself on having the best chocolate chips cookies in town. To be sure each cookie has the right number of chocolate chips, a few cookies are selected from each batch and the number of chocolate chips in each cookie is counted. This is done for several days and the following results were found: Number of Cookies Number of Chocolate Chips Identify the random variable X in this experiment and find the probability distribution of X. 2

3 We use histograms to represent the probability distributions of random variables. We place the possible values for the random variable X on the horizontal axis. We then center a bar around each x value and let its height be equal to the probability of that x value. Example 8: Referring to Example 7, a) Draw the histogram for the random variable X b) Find P(X 12) Definition: Given a sequence of n Bernoulli trials with the probability of success p the Binomial (Bernoulli) Distribution is given by P(X = r) = C(n,r)p r (1 p) (n r) for r = 0...n Example 9: A fair six-sided die is rolled 8 times. Let X represent the number of times the die lands on a number greater than 4. Find the probability distribution of X. Section 3.1 Homework Problems: 1-9 (odd), 13, 15, 19, 21, 23, 29 3

4 Section Measures of Central Tendency Example 1: Records kept by the cheif dietitian at the university caferteria over a 25-wk period show the following weekly consumption of milk (in gallons): Milk Weeks Find the average number of gallons of milk consumed per week in the cafeteria. Definition: Let X denote a random variable that assumes the values x 1,x 2,...,x n, with associated probabilites p 1, p 2,..., p n, respectively. Then the expected value of X, E(X), is given by: E(X) = x 1 p 1 + x 2 p 2 + x n p n Example 2: Referring to Example 1, let the random variable X denote the number of gallons of milk consumed in a week at the cafeteria. a) Find the probability distribution of X. b) Compute E(X) c) Draw a histogram for the random variable X Note: If we think of placing the histogram on a see-saw, the expected value occurs where we would put the fulcrum to balance it. 4

5 Example 3: In a lottery, 3000 tickets are sold for $1 each. One first prize of $5000, 1 second prize of $1000, 3 third prizes of $100, and 10 consolation prizes of $10 are to be awarded. What are the expected net earnings of a person who buys one ticket? Example 4: A woman purchased a $15,000 1-year term-life inusrance policy for $200. Assuming that the probability that she will live another year is 0.994, find the company s expected net gain. Definition: A fair game means that the expected value of the player s net winnings is zero. (E(X) = 0) Example 5: Mike and Bill play a card game with a standard deck of 52 cards. Mike selects a card from a well-shuffled deck and receives A dollars from Bill if the card selected is a diamond; otherwise, Mike pays Bill a dollar. Determine the value of A if the game is to be fair. 5

6 Example 6: If you roll a fair six-sided die 4 times, what is the expected number of times that the die will land with a two facing up? Definition: The expected value of a binomial distribution with n trials and probability of success p is Example 7: If you roll a fair six-sided die 100 times, what is the expected number of times that the die will land on a number less than three? Definition: The median is the middle value in a set of data arranged in increasing or decreasing order (when there is an odd number of entries). If there is an even number of entries, the median is the average of the two middle numbers. Definition The mode is the value that occurs most frequently in a set of data. Example 8: In each of the following cases, find the mean, median, and mode. Section 3.2 Homework Problems: 1, 5, 9, 11, 15, 17, 25, 29, 35, 37, 43 6

7 Section Measures of Spread Example 1: Draw the histograms for the random variables X and Y that have the following probability distributions: x P(X = x) y P(Y = y) Definition: Suppose a random variable X has the following probability distribution: x P(X = x) x 1 p 1 x 2 p 2 x n p n and expected value E(X) = µ. Then the variance of the random variable X is Var(X)=p 1 (x 1 µ) 2 + p 2 (x 2 µ) p n (x n µ) 2. Definition: The standard deviation of a random variable X, σ, is defined by: σ = Var(X) 7

8 Finding expected value and standard deviation using the calculator: Enter the x-values into L1 and the corresponding probabilities into L2 (STAT 1:Edit) On the homescreen type 1-Var Stats L 1, L 2 (STAT CALC 1:1-Var Stats) x is the expected value (mean) σx is the standard deviation To find variance, you would need to recall that Var(X) = σ 2 Example 2: Referring to Example 1, a) Find the variance and standard deviation of X. b) Find the variance and standard deviation of Y. Example 3: The percent of the voting age population who cast ballots in presidential election years from 1980 through 2000 are given below: Election Year Turnout Percentage Find the mean and standard deviation of the voter turnout. Example 4: If you roll a fair six-sided die 4 times, what is the variance and standard deviation of the number of times the die lands on a two? Definition: The variance of a binomial distribution with n trials and the probability of success equal to p is Example 5: If you roll a fair six-sided die 100 times, what is the variance and standard deviation of the number of times that the die will land on a number less than three? 8

9 Chebyshev s Inequality: Let X be a random variable with expected value µ and standard deviation σ. Then, the probability that a randomly chosen outcome of the experiment lies between µ kσ and µ + kσ is at least 1 1 k 2 ; that is: P(µ kσ X µ + kσ) 1 1 k 2 Example 6: A probability distribution has a mean of 10 and a standard deviation of 1.5. Use Chebyshev s inequality to estimate the probability that an outcome of the experiment lies between 7 and 13 Example 7: A probability distribution has a mean of 70 and a standard deviation of 3.2. Use Chebyshev s Inequality to find the value of c that guarantees that the probability is at least that an outcome of the experiment lies between 70 c and 70 + c. Section 3.3 Homework Problems: 1, 3, 7, 9, 13, 15, 21, 25, 33, 35, 37 9

10 Section The Normal Distribution Up until now, we have been dealing with finite discrete random variables. In finding the probability distribution, we could list the possible values in a table and represent it with a histogram. Definition: For a continuous random variable, a probability density function is defined to represent the probability distribution. Example 1: Note that the for a continous random variable, X, P(X x) = P(X < x) Definition: We concentrate on a special class of continuous probability distributions known as normal distributions. Each normal distribution is defined by µ and σ. Each normal distribution has the following characteristics: 1. The area under the curve is always The curve never crosses the x axis. 3. The peak occurs directly above µ 4. The curve is symmetric about a vertical line passing through the mean. Example 2: Definition: The standard normal variable usually denoted by Z has a normal probability distribution with µ = 0 and σ = 1. 10

11 Example 3: Find and sketch the following: a) P(Z 1.79) b) P(Z 3.49) c) P( 2 Z 1.79) Example 4: According to the data released by the Chamber of Commerce of a certain city, the weekly wages of factory workers are normally distributed with a mean of $600 and a standard deviation of $50. What is the probability that a worker selected at random from the city makes a weekly wage a) of less than $600? b) of more than $760? c) between $575 and $650? 11

12 Example 5: Let Z be the standard normal variable. Find the values of a if a satisfies: a) P(Z a) = b) P(Z a) = c) P( a Z a) = Example 6: The scores on an Econ exam were normally distributed with a mean of 72 and a standard deviation of 16. If the instructor assigns a grade of A to 8% of the class, a grade of B to 12% of the class, and a grade of C to 25% of the class, what is the lowest score a student may have and still obtain a C? Section 3.4 Homework Problems: 1, 5, 9, 13, 17, 21-33(odd) 12