Question

Suppose that the joint probability mass function for the random variables X and Y is f(x,y)=kexp(−x0​x​−y0​y​);x,y>0. Report. For generic parameters x0​,y0​,α,β :

(a) Verify that f(x,y) is a valid probability mass function.
(b) Derive expressions for the marginal distributions of g(x) and h(y).
(c) Are the random variables X and Y independent? (d) Derive the expression for the mean μX​ of X.
(e) Derive the expression for the variance σX2​ of X.
(f) Derive the expression for the probability P(x0​x​>α and y0​y​>β).

Answer 1

This answer provides a step-by-step explanation for each part of the question, including verifying the validity of the probability mass function, deriving the marginal distributions, determining independence, finding the mean and variance of X, and calculating a specific probability.

Step-by-step explanation:

(a) To verify that f(x, y) is a valid probability mass function, we need to show that it satisfies two conditions:

1. Non-negativity: f(x, y) ≥ 0 for all x, y.

2. Total probability: ∑x ∑y f(x, y) = 1.

(b) To find the marginal distributions of g(x) and h(y), we integrate f(x, y) over the other variable.

(c) To determine if X and Y are independent, we need to compare the joint probability with the product of the marginal probabilities.

(d) To find the mean μX of X, we need to calculate the expected value of X.

(e) To find the variance σX2 of X, we need to calculate the expected value of (X - μX)2.

(f) To find the probability P(x0 < X > α and y0 < Y > β), we integrate f(x, y) over the specified range.