PMF Joint and Marginal PMFs

Question 1

For two discrete random variables X and Y, the joint PMF p(x, y) must satisfy which condition?

Accepted Answer

∑_x ∑_y p(x, y) = 1 and p(x, y) ≥ 0 for all x, y

Answer

A valid joint PMF must be non-negative everywhere and sum to 1 over all possible pairs (x, y).

Question 2

The marginal PMF of X is obtained from the joint PMF p(x, y) by:

Accepted Answer

Summing p(x, y) over all values of y

Answer

The marginal PMF p_X(x) = ∑_y p(x, y) is obtained by summing the joint PMF over all values of the other variable Y.

Question 3

X and Y are discrete random variables with joint PMF p(x, y). They are independent if and only if:

Accepted Answer

p(x, y) = p_X(x) · p_Y(y) for all x, y

Answer

X and Y are independent if and only if their joint PMF factors as the product of their marginal PMFs for every pair (x, y).

Question 4

Suppose p(x, y) = c · x · y for x ∈ {1, 2} and y ∈ {1, 2}, and 0 otherwise. What is the value of c?

Accepted Answer

1/9

Answer

Summing c·x·y over all pairs: c(1·1 + 1·2 + 2·1 + 2·2) = c(1+2+2+4) = 9c = 1, so c = 1/9.

Question 5

Given joint PMF p(1,1)=0.1, p(1,2)=0.2, p(2,1)=0.3, p(2,2)=0.4, what is the marginal PMF p_X(1)?

Accepted Answer

0.3

Answer

p_X(1) = p(1,1) + p(1,2) = 0.1 + 0.2 = 0.3, summing over all values of Y.

PMF - Probability Mass Function Practice Test