Random Variables and Probability Models Lecture 8 Random Variables and Probability Models Copyright © 2012 Pearson Education. All rights reserved.

Expected Value of a Random Variable QTM1310/ Sharpe Expected Value of a Random Variable A variable whose value is based on the outcome of a random event is called a random variable. If we can list all possible outcomes, the random variable is called a discrete random variable. If a random variable can take on any value between two values, it is called a continuous random variable.

Expected Value of a Random Variable QTM1310/ Sharpe Expected Value of a Random Variable For both discrete and continuous random variables, the set of all the possible values and their associated probabilities is called the probability model. When the probability model is known, then the expected value can be calculated: (discrete random variable) is sometimes written as , but not 3

Expected Value of a Random Variable QTM1310/ Sharpe Expected Value of a Random Variable The probability model for a particular life insurance policy is shown. Find the expected annual payout on a policy. We expect that the insurance company will pay out $200 per policy per year. 4

Standard Deviation of a Random Variable QTM1310/ Sharpe Standard Deviation of a Random Variable Standard Deviation of a discrete random variable: 5

Standard Deviation of a Random Variable QTM1310/ Sharpe Standard Deviation of a Random Variable The probability model for a particular life insurance policy is shown. Find the standard deviation of the annual payout. 6

Standard Deviation of a Random Variable QTM1310/ Sharpe Standard Deviation of a Random Variable Example: Book Store Purchases Suppose the probabilities of a customer purchasing 0, 1, or 2 books at a book store are 0.2, 0.4 and 0.4 respectively. What is the expected number of books a customer will purchase? What is the standard deviation of the book purchases? 7

Standard Deviation of a Random Variable QTM1310/ Sharpe Standard Deviation of a Random Variable Example: Book Store Purchases Suppose the probabilities of a customer purchasing 0, 1, or 2 books at a book store are 0.2, 0.4 and 0.4 respectively. What is the expected number of books a customer will purchase? What is the standard deviation of the book purchases? 8

Properties of Expected Values and Variances QTM1310/ Sharpe Properties of Expected Values and Variances Adding a constant c to X: Multiplying X by a constant a: 9

Properties of Expected Values and Variances QTM1310/ Sharpe Properties of Expected Values and Variances Addition Rule for Expected Values of Random Variables Addition Rule for Variances of (independent) Random Variables 10

8.3 Properties of Expected Values and Variances QTM1310/ Sharpe 8.3 Properties of Expected Values and Variances The expected annual payout per insurance policy is $200 and the variance is $14,960,000. If the payout amounts are doubled, what are the new expected value and variance? Compare this to the expected value and variance on two independent policies at the original payout amount. Note: The expected values are the same but the variances are different. 11

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models The Uniform Model If X is a random variable with possible outcomes 1, 2, …, n and for each i, then we say X has a discrete Uniform distribution U[1, …, n]. When tossing a fair die, each number is equally likely to occur. So tossing a fair die is described by the Uniform model U[1, 2, 3, 4, 5, 6], with 12

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Bernoulli Trials Definition: A Bernoulli Trial is a trial with the following characteristics: There are only two possible outcomes (success and failure) for each trial. The probability of success, denoted p, is the same for each trial. The probability of failure is q = 1 – p. The trials are independent. The next two probability models apply to experiments with Bernoulli trials. 13

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models The Geometric Model Predicting the number of Bernoulli trials required to achieve the first success. The 10% Condition: For finite samples, the sample should be no more than 10% of the population. 14

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models The Geometric Model Find the mean (expected value) of a random variable X, using a geometric distribution with probability of success, p. E(X) = 0q + 1p = p Var(X) = (0 – p)2q + (1 – p)2p = p2q + q2p = pq(p + q) = pq(1) = pq X 1 P(X) q p 15

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Independence Bernoulli Trials must be independent. The 10% Condition: From finite populations, the sample should be no more than 10% of the population. 16

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models The Binomial Model Predicting the number of successes in a fixed number of Bernoulli trials. 17

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models The Poisson Model Predicting the number of events that occur over a given interval of time or space. The Poisson is a good model to consider whenever the data consist of counts of occurrences. 18

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models For example, a website averages 4 hits per minute. Find the probability that there will be no hits in the next minute. 19

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Example: Closing Sales A salesman normally closes a sale on 80% of his presentations. Assuming the presentations are independent, What model should be used to determine the probability that he closes his first presentation on the fourth attempt? What is the probability he closes his first presentation on the fourth attempt? 20

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Example (continued): Closing Sales A salesman normally closes a sale on 80% of his presentations. Assuming the presentations are independent, What model should be used to determine the probability that he closes his first presentation on the fourth attempt? Geometric What is the probability he closes his first presentation on the fourth attempt? 21

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Example: Professional Tennis A tennis player makes a successful first serve 67% of the time. Of the first 6 serves of the next match, What model should be used to determine the probability that all 6 first serves will be in bounds? What is the probability that all 6 first serves will be inbounds? How many first serves can be expected to be inbounds? 22

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Example (continued): Professional Tennis A tennis player makes a successful first serve 67% of the time. Of the first 6 serves of the next match, What model should be used to determine the probability that all 6 first serves will be in bounds? Binomial What is the probability that all 6 first serves will be inbounds? How many first serves can be expected to be inbounds? E(X) = np = 6(0.67) = 4.02 23

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Example: Satisfaction Survey A cable provider wants to contact customers to see if they are satisfied with a new digital TV service. If all customers are in the 452 phone exchange, (so there are 10,000 possible numbers from 452-0000 to 452-9999), What probability model would be used to model the selection of a single number? What is the probability the number selected will be an even number? What is the probability the number selected will end in 000? 24

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Example (continued): Satisfaction Survey A cable provider wants to contact customers to see if they are satisfied with a new digital TV service. If all customers are in the 452 phone exchange, (so there are 10,000 possible numbers from 452-0000 to 452-9999), What probability model would be used to model the selection of a single number? Uniform, all numbers are equally likely. What is the probability the number selected will be an even number? 0.5 What is the probability the number selected will end in 000? 0.001 25

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Example: Web Visitors A website manager has noticed that during evening hours, about 3 people per minute check out from their online shopping cart and make a purchase. She believes that each purchase is independent of the others. What probability model would be used to model the number of purchases per minute? What is the probability that in any one minute at least one purchase is made? What is the probability that no one makes a purchase in the next 2 minutes? 26

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Example (continued): Web Visitors A website manager has noticed that during evening hours, about 3 people per minute check out from their online shopping cart and make a purchase. She believes that each purchase is independent of the others. What probability model would be used to model the number of purchases per minute? The Poisson What is the probability that in any one minute at least one purchase is made? 0.9502 27

8.4 Discrete Probability Models QTM1310/ Sharpe 8.4 Discrete Probability Models Example (continued): Web Visitors A website manager has noticed that during evening hours, about 3 people per minute check out from their online shopping cart and make a purchase. She believes that each purchase is independent of the others. What is the probability that no one makes a purchase in the next 2 minutes? 28

Probability distributions are still just models. QTM1310/ Sharpe Probability distributions are still just models. • If the model is wrong, so is everything else. Watch out for variables that aren’t independent. Don’t write independent instances of a random variable with notation that looks like they are the same variables. 29

• Be sure you have Bernoulli trials. QTM1310/ Sharpe Don’t forget: Variances of independent random variables add. Standard deviations don’t. • Don’t forget: Variances of independent random variables add, even when you’re looking at the difference between them. • Be sure you have Bernoulli trials. 30

QTM1310/ Sharpe What Have We Learned? Understand how probability models relate values to probabilities. • For discrete random variables, probability models assign a probability to each possible outcome. Know how to find the mean, or expected value, of a discrete probability model and the standard deviation. 31

QTM1310/ Sharpe What Have We Learned? Foresee the consequences of shifting and scaling random variables, specifically understand the Law of Large Numbers and that the common understanding of the “Law of Averages” is false. E(X ± c) = E(X) ± c E(aX) = aE(X) Var(X ± c) = Var(X) Var(aX) = a2Var(X) SD(X ± c) = SD(X) SD(aX) = |a|SD(X) 32

QTM1310/ Sharpe What Have We Learned? Understand that when adding or subtracting random variables the expected values add or subtract well: E(X ± Y) = E(X) ± E(Y). However, when adding or subtracting independent random variables, the variances add: Var(X ±Y) = Var(X) + Var(Y) Be able to explain the properties and parameters of the Uniform, the Binomial, the Geometric, and the Poisson distributions. 33