Section Distributions of Random Variables

Transcription

1 Section 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 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. A 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 an infinite number of possible values. (i.e. usually measurements) Example 4: Referring to Example 1, find the probability distribution of the random variable X. 1

2 Example 5: The number of cars waiting in line at the beginning of 2-minute intervals at a certain bank was observed. The following data was collected: Number of cars Frequency of occurence Let X represent the number of cars observed waiting in line and find the probability distribution of X 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 6: Referring to Example 5, a) Draw the histogram for the random variable X b) Find P (X 2) 2

3 Section Expected Value 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. 3

4 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? Definition: A fair game means that the expected value of the player s net winnings is zero. (E(X) = 0) Example 4: 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. 4

5 Example 5: 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: If P (E) is the probability of an event E occuring, then a) The odds in favor of E occuring are P (E) 1 P (E) = P (E) P (E c ) [P (E) 1] b) The odds against E occuring are 1 P (E) P (E) = P (Ec ) P (E) [P (E) 0] Example 6: The probability of an event E occurring is.8. a) What are the odds in favor of E occurring? b) What are the odds against E occurring? Definition: If the odds in favor of an event E occuring are a to b, then the probability of E occurring is P (E) = a a + b Example 7: If a sports forecaster states that the odds of a certain boxer winning a match are 4 to 3, what is the probability that the boxer will win the match? 5

6 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. 6

7 Section Variance and Standard Deviation 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 = V ar(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 V ar(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. Chebychev 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 3: A probability distribution has a mean of 10 and a standard deviation of 1.5. Use Chebychev s inequality to estimate the probability that an outcome of the experiment lies between 7 and 13 8

9 Section The Binomial Distribution Binomial Experiments have the following properties: 1. The number of trials in the experiment is fixed. 2. There are 2 possible outcomes in the experiment: success and failure 3. The probability of success in each trial is the same. 4. Trials are independent of each other. Calculating a Binomial Probability: 1. Determine if the experiment is binomial. 2. Determine the number of trials (n). 3. Define success in the experiment and determine the probability of that success occuring (p). 4. Determine the number of successes desired (r). 5. Calculate the desired probability: (a) By hand, the probability is found by doing the following calculation: P (X = r) = C(n, r)p r (1 p) (n r) (b) We can use the calculator program BINOM to do these calculations for us. Example 1: What is the probability of exactly 3 successes in 6 trials of a binomial experiment in which p = 1/2? Example 2: A fair die is cast 4 times. Compute the probability of obtaining exactly one 6 in the four throws. 9

10 Example 3: A biology quiz consists of 8 multiple choice questions. Five must be answered correctly to receive a passing grade. If each question has 5 possible answers of which only one is correct, what is the probability that a student who guesses at random on each question will pass the examination? Definition: For a binomial random variable X, we have the following: E(X) = np Var(X) = np(1 p) σ = np(1 p) Example 4: At a certain university the probability that an entering freshman will graduate within 4 years is.6. From an incoming class of 2000 freshman, find a) The expected number of students who will graduate within 4 years. b) The standard deviation of the number of students who will graduate within 4 years. 10

11 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. 11

12 Example 3: Find and sketch the following: a) P (Z 1.79) b) P (Z 3.49) c) P ( 2 Z 1.79) Suppose the random variable X has a normal distribution with mean µ and standard deviation σ. We can calculate probabilities for X by converting the normal distribution to a standard normal distribution. We do this by converting the bounds of X to bounds of Z using the following formula: z = x µ σ Example 4: Suppose X is a normal random variable with µ = 500 and σ = 75. Find P (400 X 600) Example 5: Let Z be the standard normal variable. Find the values of z if z satisfies: a) P (Z z) =.8907 b) P ( z Z z) =

13 Section Applications of the Normal Distribution Example 1: 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 $550 and $650? Example 2: 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 15% of the class, what is the lowest score a student may have and still obtain an A? 13

14 Example 3: Let s look at the following example. Suppose a fair coin is tossed 5 times. Let X represent the number of times the coin lands on heads. X has a binomial distribution. a) Find the probability distribution for X. b) Draw the histogram for X. c) Find P (X 3) d) Use the appropriate normal distribution to approximate the binomial probability in part c. 14

15 Steps to approximating binomial probabilities with normal distributions: 1. Identify n and p. 2. Calculate µ and σ: µ = np σ = np(1 p) 3. Convert the binomial random variable X to the normal random variable Y : Draw a rough sketch of the histogram of X and the normal curve of Y. Determine which rectangles you are wanting to include. Find the appropriate interval for the area under the normal curve by adding or subtracting 0.5 to your bounds. 4. Compute the desired area under the normal curve using normalcdf. Example 4: A basketball player has a 60% chance of making a free throw. What is the probability of her making 250 or more free throws in 400 trials? Use the appropriate normal distribution to approximate the binomial distribution. 15