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

Covariance

Joint and marginal PMFs of the discrete r.v. X (Girls) and Y (Boys) for family example Boys, B Total Girls, G Total

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 f X (x) = 12x (1 – x) 2, f Y (y) = 12y (1 – y) 2 What is the Cov(X,Y)?

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

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

Example 7.20: Correlation (2) Suppose that A and B are events with positive probability. Show that I A and I B are positively correlated, negatively correlated, or uncorrelated depending on whether P(B|A) is greater than, less than, or equal to P(B). x

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 f X (x) = 12x (1 – x) 2, f Y (y) = 12y (1 – y) 2, Cov(X,Y)= What is the  (X,Y)?

Example: Correlation (4) a) Let X be uniformly distributed over (0,1) and Y= X 2. Find  (X,Y). b) Let X be uniformly distributed over (-1,1) and Y = X 2. Find Cov (X,Y).

Table : Conditional PMF of Y (Boys) for each possible value of X (Girls) Boys, B p X (x) Girls, G p Y (y) Determine and interpret the conditional expectation of the number of boys given the number of girls is 2?

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 f X (x) = 12x (1 – x) 2, f Y (y) = 12y (1 – y) 2 What is the conditional expectation of Y given X = x?

Example 7: Law of Total Expectation Supposed that we know that the average height of 2 men is 6’ = 72” and the average height of 3 women is 5’5” = 65”. a) Determine the mean height of all of the people. b) Formulate the work in part (a) in ‘probability’. x

Table : Conditional PMF of Y (Boys) for each possible value of X (Girls) Boys, B p X (x) Girls, G p Y (y)

Example: Double Expectation (1) Suppose that N(t), the number of people who pass by a museum at or prior to t, is a Poisson process having rate λ. If a person passing by enters the museum with probability p, what is the expected number of people who enter the museum at or prior to t? x

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.

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 f X (x) = 12x (1 – x) 2, f Y (y) = 12y (1 – y) 2 What is the conditional variance of Y given X = x?

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.