Question

A) Determine whether the following function is a valid probability density function (PDF) for a company that manufactures screws of at most 2 inches in length:

Given the joint distribution f(x, y) defined as:

f(x, y) = (7/6)(y^2 + 2xy), if 0 <= x <= 2, 0 <= y <= 1

Please verify if this is a proper probability density function.

b) Calculate the marginal density function of Y, denoted as f(Y).

c) Determine the constant 'c' such that P(0.2 < Y < c) = 0.5.

d) Calculate the expected value E[X] and the variance Var(X).

e) Calculate the expected value E[Y] and the variance Var(Y).

f) Find the covariance between X and Y.

g) Calculate E[Y|X=x] and Var(Y|X=x) for a given value of X.

Answer 1

To determine if the given function is a valid probability density function (PDF), several conditions need to be checked. One condition is that the function must be non-negative for all values of x and y in the given range. Additionally, the function must integrate to 1 over the entire range.

Step-by-step explanation:

To determine if the given function is a valid probability density function (PDF), several conditions need to be checked. One condition is that the function must be non-negative for all values of x and y in the given range. Additionally, the function must integrate to 1 over the entire range.

To check the first condition, we need to ensure that the function is non-negative. Evaluating the function f(x, y) = (7/6)(y^2 + 2xy) for all values of x and y in the given range, we find that it is indeed non-negative.

To check the second condition, we need to evaluate the integral of the function f(x, y) over the entire range. Integrating f(x, y) with respect to y from 0 to 1, and then integrating the result with respect to x from 0 to 2, we find that the integral evaluates to 1. Therefore, the given function is a valid PDF for a company that manufactures screws of at most 2 inches in length.