Question

Consider the set E = . We choose a point (X, Y) uniformly at random in E, and the joint PDF of X and Y is given by:

f_XY(x, y) =
{
C, for (x, y) ∈ E,
0, otherwise
}

(a) Find the constant C.

(b) Find the marginal PDFs f_X(x) and f_Y(y).

(c) Are X and Y independent?

Answer 1

The constant C for the joint PDF is 1/2. The marginal PDFs require integration over the variable not being considered, and they depend on the limits set by E. X and Y are not independent because the limit on one variable affects the possible values of the other within set E.

Step-by-step explanation:

For the given set E = , we are tasked with finding the uniform joint PDF fXY(x, y) and checking the independence of X and Y.

Finding Constant C

The constant C can be found by insuring that the total probability over the region E is equal to 1. Since the probability density function is uniform over E, we can find C by calculating the area of E and setting this equal to 1. The area of E, being the diamond shape formed by the constraints |x| + |y| ≤ 1, is 2. Hence,

C × area of E = 1 → C × 2 = 1 → C = 1/2.

Finding Marginal PDFs fX(x) and fY(y)

The marginal PDFs can be found by integrating the joint PDF over the other variable. For fX(x), integrate fXY over y from -1 to 1, taking into account the shape of E. Similarly, for fY(y), integrate over x.

Checking Independence of X and Y

X and Y are not independent because the value of one affects the possible values of the other given the constraints of set E. For example, if X is near its maximum value of 1, Y must be near 0, which shows a dependency between the two variables.