Chapters 7 and 10: Expected Values of Two or More Random Variables

1. Chapters 7 and 10: Expected Values of Two or More Random Variables http://blogs.oregonstate.edu/programevaluation/2011/02/18/timely-topic-thinking-carefully/

2. Covariance

3. Joint and marginal PMFs of the discrete r.v. X (Girls) and Y (Boys) for family example

4. Example: Covariance(1) A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. The joint PDF is fX(x) = 12x (1 – x)2, fY(y) = 12y(1 – y)2 What is the Cov(X,Y)?

5. Example: Covariance (2) a) Let X be uniformly distributed over (0,1) and Y= X2. Find Cov (X,Y). b) Let X be uniformly distributed over (-1,1) and Y = X2. Find Cov (X,Y).

6. Example: Correlation (1) a) Let X be uniformly distributed over (0,1) and Y= X2. Find Cov (X,Y). b) Let X be uniformly distributed over (-1,1) and Y = X2. Find Cov (X,Y).

7. Example: Correlation (3) A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. The joint PDF is fX(x) = 12x (1 – x)2, fY(y) = 12y(1 – y)2, Cov(X,Y)=-0.0267 What is the (X,Y)?

8. Example: Correlation (4) a) Let X be uniformly distributed over (0,1) and Y= X2. Find (X,Y). b) Let X be uniformly distributed over (-1,1) and Y = X2. Find Cov (X,Y).

9. Table : Conditional PMF of Y (Boys) for each possible value of X (Girls) Determine and interpret the conditional expectation of the number of boys given the number of girls is 2?

10. Example: Conditional Expectation A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. The joint PDF is fX(x) = 12x (1 – x)2, fY(y) = 12y(1 – y)2 What is the conditional expectation of Y given X = x?

11. Table : Conditional PMF of Y (Boys) for each possible value of X (Girls)

12. Example: Double Expectation (2) A quality control plan for an assembly line involves sampling n finished items per day and counting X, the number of defective items. Let p denote the probability of observing a defective item. p varies from day to day and is assume to have a uniform distribution in the interval from 0 to ¼. a) Find the expected value of X for any given day.

13. Example: Conditional Variance A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. The joint PDF is fX(x) = 12x (1 – x)2, fY(y) = 12y(1 – y)2 What is the conditional variance of Y given X = x?

14. Example: Law of Total Variance A fisherman catches fish in a large lake with lots of fish at a Poisson rate (Poisson process) of two per hour. If, on a given day, the fisherman spends randomly anywhere between 3 and 8 hours fishing, find the expected value and variance of the number of fish he catches.