Question

A car dealership sells 0, 1, or 2 luxury cars on any day. When selling a...

A car dealership sells 0, 1, or 2 luxury cars on any day. When selling a car, the dealer also tries to persuade the customer to buy an extended warranty for the car. Let X denote the number of luxury cars sold in a given day, and let Y denote the number of extended warranties sold.

P(X = 0,Y = 0) = 1/6;

P(X = 1,Y = 0) = 1/12;

P(X = 1,Y = 1) = 1/6;

P(X = 2,Y = 0) = 1/12;

P(X = 2,Y = 1) = 1/3;

P(X = 2,Y = 2) = 1/6.

(a) Find the marginal distributions of X and Y; (Drawing the table of the joint distribution might be helpful.)
(b) Calculate conditional distribution p (x | y = 0) and E (X | Y = 0);
(c) Calculate Cov(X, Y ) and the correlation of X and Y, ρXY. Are X and Y independent random variables? Explain the reason.

(d) Calculate Va r (X + Y ).

a) The marginal distributions are,

b) The conditional distribution is

The conditional expectation is

c) The Covariance is found as

Since , X, Y are not independent.

d) The variance

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