Chapter 16. Random Variables. Copyright 2010, 2007, 2004 Pearson Education, Inc.

Expected Value: Center A random variable is a numeric value based on the outcome of a random event. We use a capital letter, like X, to denote a random variable. A particular value of a random variable will be denoted with the corresponding lower case letter, in this case x. Ex. X = The amount a company pays out on an individual insurance policy x = $10,000 if you die that year x = $5,000 if you were disabled x = $0 if neither occur Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-3

Expected Value: Center (cont.) There are two types of random variables: Discrete random variables can take one of a countable number of distinct outcomes. Example: Number of credit hours Continuous random variables can take any numeric value within a range of values. Example: Cost of books this term Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-4

Discrete or Continuous? 1. Time you listen to the radio. 2. Number of pets in one house. 3. Length of a leaf in your yard. 4. Width of your yard. 5. Amount of siblings. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-5

Expected Value: Center (cont.) A probability model for a random variable consists of: The collection of all possible values of a random variable, and the probabilities that the values occur. Of particular interest is the value we expect a random variable to take on, notated μ (for population mean) or E(X) for expected value. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-6

Expected Value: Center (cont.) The expected value of a (discrete) random variable can be found by summing the products of each possible value by the probability that it occurs: E X x P x Note: Be sure that every possible outcome is included in the sum and verify that you have a valid probability model to start with. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-7

First Center, Now Spread For data, we calculated the standard deviation by first computing the deviation from the mean and squaring it. We do that with discrete random variables as well. The variance for a random variable is: 2 2 Var X x P x The standard deviation for a random variable is: SD X Var X Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-8

Example (p. 383 #2, 10) Find the expected value and standard deviation of each random variable. a) X 0 1 2 P(X=x) 0.2 0.4 0.4 b) X 100 200 300 400 P(X=x) 0.1 0.2 0.5 0.2 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-9

Example (p. 383 #4, 12) You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose. a) Create a probability model for the amount you win. b) Find the expected amount you ll win. c) What would you be willing to pay to play this game? Explain d) Find the standard deviation of the amount you might win rolling a die. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-10

Example (p. 384 #22) Your company bids for two contracts. You believe the probability you get contract #1 is 0.8. If you get contract #1, the probability you also get contract #2 will be 0.2, and if you do not get contract #1, the probability you get contract #2 will be 0.3 a) Are the two contracts independent? Explain. b) What s the probability you get both contracts? c) What s the probability you get no contracts? d) Let random variable X be the number of contracts you get. Find the probability model for X. e) What are the expected value and standard deviation? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-11

More About Means and Variances Adding or subtracting a constant from data shifts the mean but doesn t change the variance or standard deviation: E(X ± c) = E(X) ± c Var(X ± c) = Var(X) Example: Consider everyone in a company receiving a $5000 increase in salary. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-13

More About Means and Variances (cont.) In general, multiplying each value of a random variable by a constant multiplies the mean by that constant and the variance by the square of the constant: E(aX) = ae(x) Var(aX) = a 2 Var(X) Example: Consider everyone in a company receiving a 10% increase in salary. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-14

More About Means and Variances (cont.) In general, The mean of the sum of two random variables is the sum of the means. The mean of the difference of two random variables is the difference of the means. E(X ± Y) = E(X) ± E(Y) If the random variables are independent, the variance of their sum or difference is always the sum of the variances. Var(X ± Y) = Var(X) + Var(Y) Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-15

Example (p. 384 #28) Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of: a) 2Y + 20 b) 3X c) 0.25X + Y d) X 5Y e) X 1 + X 2 + X 3 Mean SD X 80 12 Y 12 3 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-16

Continuous Random Variables Random variables that can take on any value in a range of values are called continuous random variables. Now, any single value won t have a probability, but Continuous random variables have means (expected values) and variances. We won t worry about how to calculate these means and variances in this course, but we can still work with models for continuous random variables when we re given the parameters. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-17

Continuous Random Variables (cont.) Good news: nearly everything we ve said about how discrete random variables behave is true of continuous random variables, as well. When two independent continuous random variables have Normal models, so does their sum or difference. This fact will let us apply our knowledge of Normal probabilities to questions about the sum or difference of independent random variables. Back to z-scores! Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-18

Example (p. 385 #38) The American Veterinary Association claims that the annual cost of medical care for dogs averages $100, with a standard deviation of $30, and for cats averages $120, with a standard deviation of $35. a) What s the expected difference in the cost of medical care for dogs and cats? b) What s the standard deviation of this difference? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-20

Example (p. 385 #38) The American Veterinary Association claims that the annual cost of medical care for dogs averages $100, with a standard deviation of $30, and for cats averages $120, with a standard deviation of $35. c) If the costs can be described by Normal models, what s the probability that medical expenses are higher for someone s dog that for her cat? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-21

Example (p. 385 #40) You re thinking about getting two dogs and a cat. Assume that annual veterinary expenses are independent and have a Normal model with the means and standard deviations from Exercise 38. a) Define appropriate variables and express the total annual veterinary costs you may have. b) Describe the model for this total cost. Be sure to specify its name, expected value, and standard deviation. c) What s the probability that your total expenses will exceed $400? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-22

What Can Go Wrong? Probability models are still just models. Models can be useful, but they are not reality. Question probabilities as you would data, and think about the assumptions behind your models. If the model is wrong, so is everything else. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-23

What Can Go Wrong? (cont.) Don t assume everything s Normal. You must Think about whether the Normality Assumption is justified. Watch out for variables that aren t independent: You can add expected values for any two random variables, but you can only add variances of independent random variables. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-24

What Can Go Wrong? (cont.) 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. Don t write independent instances of a random variable with notation that looks like they are the same variables. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-25

What have we learned? We know how to work with random variables. We can use a probability model for a discrete random variable to find its expected value and standard deviation. The mean of the sum or difference of two random variables, discrete or continuous, is just the sum or difference of their means. And, for independent random variables, the variance of their sum or difference is always the sum of their variances. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-26

What have we learned? (cont.) Normal models are once again special. Sums or differences of Normally distributed random variables also follow Normal models. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 16-27