2. Copyright © 2009 Pearson Education, Inc. Chapter 16 Random Variables

3. Copyright © 2009 Pearson Education, Inc. Slide 1- 3 Objectives: The student will be able to: Recognize random variables. Find the probability model for a discrete random variable. Find and interpret in context the mean (expected value) and the standard deviation of a random variable.

4. Copyright © 2009 Pearson Education, Inc. Slide 1- 4 Random Variables A random variable assumes a 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 a lower case letter, in this case x. Examples: Let X = the roll of a die Let X = five times the roll of a die Let X = number of rolls of a die until you roll a 6 Let X= the sum of 3 rolls of a die We might ask what is the P(X=5) for example

5. Copyright © 2009 Pearson Education, Inc. Slide 1- 5 Random Variables There are two types of random variables: Discrete random variables can take one of a finite 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

6. Copyright © 2009 Pearson Education, Inc. Slide 1- 6 Probability Models for Random Variables 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. We’ll make a table x, f(x), P(x) Let X = the roll of a die Let X= the sum of 2 rolls of a die Let X = five times the roll of a die Let X = number of rolls of a die until you roll a 6

7. Copyright © 2009 Pearson Education, Inc. Slide 1- 7 Expected Value: Center We can describe these models using the methods we’ve seen before. We can look at the histogram for discrete random variables or a graph for a continuous random variable We are particularly interested in the value we expect a random variable to take on. We’ll denote this using μ (for population mean) or E(X) for expected value. Lets look at the probability model for X = the sum of rolling two dice. We need a table of x, f(x), and P(x),

8. Copyright © 2009 Pearson Education, Inc. Slide 1- 8 Expected Value: Center (cont.) The expected value of a (discrete) random variable can be found by summing the products of each possible value and the probability that it occurs: Note: Be sure that every possible outcome is included in the sum and verify that you have a valid probability model to start with. Compute E(X) for X = the sum of rolling two dice.

9. Copyright © 2009 Pearson Education, Inc. Example (from text) A restaurant gives every couple an opportunity to win a discount. The waiter brings 4 aces to the table. If the couple picks a black ace, they owe the full amount. If they pick the ace of hearts, they’ll get a $20 discount. If they pick the ace of diamonds, they get to pick one of the remaining cards and if it’s the ace of hearts, they’ll get a $10 discount. What is the expected discount for a couple? We need to make a table: Outcome, x, P(X=x) E(x) = Σ x*P(x) Slide 1- 9

10. Copyright © 2009 Pearson Education, Inc. Slide 1- 10 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: The standard deviation for a random variable is: Compute the standard deviation for the sum of rolling two dice. Do the same for the restaurant example

11. Copyright © 2009 Pearson Education, Inc. Slide 1- 11 Using the TI-83 Much of what you have to do for discrete random data is very much like what we did to get the descriptive statistics for data involving frequencies only here we will use decimals, probabilities. Example: A recent study involving households and the number of children in the household showed that 18% of the households had no children, 39% had one, 24% had two, 14% had three, 4% has four and 1% had five. Find the expected value, mean number of children per household, and the standard deviation of this distribution. Place the values of the variable in L1: 0,1,2,3,4,5 and the associated probabilities in L2:.18,.39,.24,.14,.04 and.01.

12. Copyright © 2009 Pearson Education, Inc. Slide 1- 12 Using the TI-83 Then do Stat -> Calc -> 1-var stat(L1, L2) The mean is 1.5 which is also the sum…. the calculator is doing what we would do if we used the formula… multiply each value by its corresponding probability and then add the separate products. The standard deviation is 1.1180. Notice the n=1. That just means that the individual probabilities added up to one which they should if this is a proper distribution. If you need to see the probability histogram, think about the classes and the boundaries: -.5 to.5;.5 to 1.5 etc. The y values are the probabilities which go no lower than 0 and no higher than.39. So here is a useable window: xmin -.5 xmax 5.5 xscl 1 ymin 0 ymax.39 or a little higher if you don't want the bars to go to the top of the screen leave yscl and xres alone. Hit the graph button.

13. Copyright © 2009 Pearson Education, Inc. Slide 1- 13 Practice 4) A day trader buys an option on a stock that will return $100 profit if it goes up today and loses $200 if it goes down. If the trader thinks there is a 75% chance that the stock will go up, what is his expected value of the option? 6,14) 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. Create a probability model for this game Find the expected amount you’ll win How much would you be willing to pay to play this game? Find the standard deviation of the amount you might win

14. Copyright © 2009 Pearson Education, Inc. 24) Your company bids for two contracts. You believe the probability that you get contract #1 is 0.8. If you get contract #1, the probability that you get contract #2 is 0.2, and if you do not get contract #1, the probability that you get #2 will be 0.3. a) Are the two contracts independent? Explain b) Find the probability that you get both contracts c) Find the probability that you get no contract d) Let X be the number of contracts you get. Find the probability model for X e) Find the expected value and standard deviation of X Slide 1- 14

15. Copyright © 2009 Pearson Education, Inc. Stop! The rest of the slides are FYI… and will not be on the exam Slide 1- 15

16. Copyright © 2009 Pearson Education, Inc. Slide 1- 16 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. Suppose we adjust both our dice to make the numbers run 2 through 7 instead of 1 through 6, how would this affect the mean and standard deviation of the sum of rolling both dice?

17. Copyright © 2009 Pearson Education, Inc. Slide 1- 17 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.

18. Copyright © 2009 Pearson Education, Inc. Slide 1- 18 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)

19. Copyright © 2009 Pearson Education, Inc. Slide 1- 19 Combining Random Variables (The Bad News) It would be nice if we could go directly from models of each random variable to a model for their sum. But, the probability model for the sum of two random variables is not necessarily the same as the model we started with even when the variables are independent. Thus, even though expected values may add, the probability model itself is different.

20. Copyright © 2009 Pearson Education, Inc. Slide 1- 20 Continuous Random Variables Random variables that can take on any value in a range of values are called continuous random variables. 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 these parameters.

21. Copyright © 2009 Pearson Education, Inc. Slide 1- 21 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. Combining Random Variables (The Good News)