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QTM1310/ Sharpe 7.1 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. Chapter 7 Random Variables and Probability Models Copyright © 2015 Pearson Education. All rights reserved. 7-1 7.1 Expected Value of a Random Variable 7-4 Copyright © 2015 Pearson Education. All rights reserved. 7.1 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. A variable whose value is based on the outcome of a random event is called a random variable. 1 2 997 $50, 000 $0 1000 1000 1000 E X $100, 000 If we can list all possible outcomes, the random variable is called a discrete random variable. $200 If a random variable can take on any value between two values, it is called a continuous random variable. Copyright © 2015 Pearson Education. All rights reserved. We expect that the insurance company will pay out $200 per policy per year. 7-2 7-5 Copyright © 2015 Pearson Education. All rights reserved. 7.2 Standard Deviation of a Random Variable 7.1 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. Standard Deviation of a discrete random variable: 2 Var X x P x 2 SD X Var X When the probability model is known, then the expected value can be calculated: E X x P x (discrete random variable) E X can be written as but never as x or y Copyright © 2015 Pearson Education. All rights reserved. 7-3 Copyright © 2015 Pearson Education. All rights reserved. 7-6 1 QTM1310/ Sharpe 7.2 Standard Deviation of a Random Variable 7.3 Properties of Expected Values and Variances The probability model for a particular life insurance policy is shown. Find the standard deviation of the annual payout. Adding a constant c to X: Multiplying X by a constant a: Copyright © 2015 Pearson Education. All rights reserved. 7-7 Copyright © 2015 Pearson Education. All rights reserved. 7.2 Standard Deviation of a Random Variable 7.3 Properties of Expected Values and Variances 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. Addition Rule for Expected Values of Random Variables What is the expected number of books a customer will purchase? 7-10 Addition Rule for Variances of (independent) Random Variables What is the standard deviation of the book purchases? Copyright © 2015 Pearson Education. All rights reserved. 7-8 Copyright © 2015 Pearson Education. All rights reserved. 7.2 Standard Deviation of a Random Variable 7.3 Properties of Expected Values and Variances Example: Book Store Purchases The expected annual payout per insurance policy is $200 and the variance is $14,960,000. 0 0.2 1 0.4 2 0.4 7-11 If the payout amounts are doubled, what are the new expected value and variance? 1.2 2 x P x 2 E 2 X 2 E X 2 200 $400 = (0 - 1.2) (0.2) (1 - 1.2) 2 (0.4) (2 - 1.2) 2 (0.4) = 0.288 + 0.016 + 0.256 = 0.56 2 Var 2 X 22 Var X 4 14,960,000 59,840,000 = 0.56 0.748 Copyright © 2015 Pearson Education. All rights reserved. 7-9 Copyright © 2015 Pearson Education. All rights reserved. 7-12 2 QTM1310/ Sharpe 7.3 Properties of Expected Values and Variances 7.4 Bernoulli Trials Compare this to the expected value and variance on two independent policies at the original payout amount. Bernoulli trials have the following characteristics: • There are only two outcomes per trial, called success and failure. • The probability of success, called p, is the same on every trial (the probability of failure, 1 – p, is often called q). E X Y E X E Y 2 200 $400 Var X Y Var X Var Y 2 14,960,000 29,920,00 • The trials are independent. Note: The expected values are the same but the variances are different. 7-13 Copyright © 2015 Pearson Education. All rights reserved. Copyright © 2015 Pearson Education. All rights reserved. 7.3 Properties of Expected Values and Variances 7.4 Bernoulli Trials The association of two random variables can be measured using the covariance of X and Y: Examples of Bernoulli trials include tossing a coin, collecting yes / no responses from a survey, or shooting free throws in basketball. Cov X , Y E X Y When using Bernoulli trials to develop probability models, we require that trial are independent. Then, the covariance gives us the extra information needed to find the variance of the sum or difference of two random variables when they are not independent: We can check the 10% Condition: As long as the number of trials or sample size is less than 10% of the population size, we can proceed with confidence that the trials are independent. Var X Y Var X Var Y 2Cov X , Y 7-14 Copyright © 2015 Pearson Education. All rights reserved. 7.3 Properties of Expected Values and Variances Copyright © 2015 Pearson Education. All rights reserved. Copyright © 2015 Pearson Education. All rights reserved. 7-17 7.5 Discrete Probability Models The Uniform Model If X is a random variable with possible outcomes 1, 2, …, n and P X i 1/ n for each i, then we say X has a discrete Uniform distribution U[1, …, n]. Covariance doesn’t have to be between –1 and +1, which makes it harder to interpret. To fix this “problem”, we can divide the covariance by each of the standard deviations to get the correlation: Corr X , Y 7-16 Cov X , Y 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 P X i 1/ 6. XY 7-15 Copyright © 2015 Pearson Education. All rights reserved. 7-18 3 QTM1310/ Sharpe 7.5 Discrete Probability Models 7.5 Discrete Probability Models The Geometric Model Predicting the number of Bernoulli trials required to achieve the first success. 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? Copyright © 2015 Pearson Education. All rights reserved. 7-19 7.5 Discrete Probability Models Copyright © 2015 Pearson Education. All rights reserved. 7-22 7.5 Discrete Probability Models The Binomial Model Predicting the number of successes in a fixed number of Bernoulli trials. Example (continued): Closing Sales 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? P( X 4) (0.20)3 (0.80) 0.0064 Copyright © 2015 Pearson Education. All rights reserved. 7-20 7.5 Discrete Probability Models Copyright © 2015 Pearson Education. All rights reserved. 7-23 7.5 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. 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 in bounds? How many first serves can be expected to be in bounds? Copyright © 2015 Pearson Education. All rights reserved. 7-21 Copyright © 2015 Pearson Education. All rights reserved. 7-24 4 QTM1310/ Sharpe 7.5 Discrete Probability Models 7.5 Discrete Probability Models Example (continued): Professional Tennis 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 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? What probability model would be used to model the number of purchases per minute? 6 P( X 6) (0.67)6 (0.33)0 0.0905 6 What is the probability that in any one minute at least one purchase is made? How many first serves can be expected to be inbounds? E(X) = np = 6(0.67) = 4.02 What is the probability that no one makes a purchase in the next 2 minutes? Copyright © 2015 Pearson Education. All rights reserved. 7-25 7.5 Discrete Probability Models 7-28 Copyright © 2015 Pearson Education. All rights reserved. 7.5 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? Example (continued): Web Visitors 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? P( X 1) 1 P( X 0) 1 e 3 30 1 0.0498 0.9502 0! What is the probability the number selected will end in 000? Copyright © 2015 Pearson Education. All rights reserved. 7-26 7.5 Discrete Probability Models 7.5 Discrete Probability Models Example (continued): Satisfaction Survey Example (continued): Web Visitors What probability model would be used to model the selection of a single number? Uniform, all numbers are equally likely. What is the probability that no one makes a purchase in the next 2 minutes? What is the probability the number selected will be an even number? 0.5 P ( X 0) What is the probability the number selected will end in 000? 0.001 Copyright © 2015 Pearson Education. All rights reserved. 7-29 Copyright © 2015 Pearson Education. All rights reserved. 7-27 Copyright © 2015 Pearson Education. All rights reserved. e 6 30 0.00248 0! 7-30 5 QTM1310/ Sharpe What Have We Learned? • Probability distributions are still just models. Foresee the consequences of shifting and scaling random variables: • If the model is wrong, so is everything else. 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) • 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. 7-31 Copyright © 2015 Pearson Education. All rights reserved. Copyright © 2015 Pearson Education. All rights reserved. 7-34 What Have We Learned? • 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. 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: • Be sure you have Bernoulli trials. 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. 7-32 Copyright © 2015 Pearson Education. All rights reserved. Copyright © 2015 Pearson Education. All rights reserved. 7-35 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: E X x P x Copyright © 2015 Pearson Education. All rights reserved. = x P x 2 7-33 6