Question

"Given the joint density of random variables X and Y as f(x,y) = c(x+y) for 0 < x, y < 2, and otherwise, you are asked to perform various tasks:

(a) Find the value of the constant c.
(b) Calculate the probability P(Y > X + 1).
(c) Determine the density of X.
(d) Discuss whether X and Y are identically distributed and provide an explanation.
(e) Discuss whether X and Y are independent and provide an explanation.
(f) Find the expected value E(XY).

Answer 1

The student's problem involves finding a constant in a joint probability density function, calculating a specific probability, and determining densities, expected values, and whether two random variables are identically distributed or independent. Essential steps include integrating over appropriate bounds and comparing resulting functions.

Step-by-step explanation:

Finding the Constant c and Probability Tasks

To solve the given problem, we must first find the value of the constant c for the joint density function f(x, y) = c(x + y) in the domain 0 < x, y < 2.

(a) To find c, we must ensure that the integral of f(x, y) over the specified domain is equal to 1:

1. Integrate f(x, y) over the domain for both x and y to get the total probability.
2. Set this integral equal to 1 and solve for c.

(b) To calculate P(Y > X + 1), we:

1. Set up the integral of f(x, y) over the region where y > x + 1 within the given domain.
2. Calculate the integral to obtain the probability.

(c) The density of X, fX(x), is found by:

1. Integrating f(x, y) with respect to y over the domain 0 < y < 2.
2. This will yield the marginal probability density function of X.

(d) To determine if X and Y are identically distributed, we:

1. Compare the marginal distributions fX(x) and fY(y).
2. If they have the same distribution, they are identically distributed.

(e) To discuss if X and Y are independent, we:

1. Check if the product of the marginal densities fX(x) * fY(y) equals f(x, y).
2. If it does, X and Y are independent; otherwise, they are not.

(f) Finally, to find the expected value E(XY), we:

1. Integrate the product x * y * f(x, y) over the domain for both x and y.
2. Calculate this integral, which will yield the expected value of XY.

Note:

The integration bounds and constraints must be taken into account at each step to ensure appropriate limits of integration.