PMF Joint and Marginal PMFs 2

Question 1

If X and Y are independent discrete random variables, what is E[XY]?

Accepted Answer

E[X] · E[Y]

Answer

For independent random variables, E[XY] = E[X] · E[Y] because the joint PMF factors into the product of marginals.

Question 2

Given p(x, y) for x ∈ {0, 1} and y ∈ {0, 1, 2}, how many distinct probability values define the joint PMF?

Accepted Answer

6

Answer

The joint PMF is defined over 2 × 3 = 6 pairs (x, y), each requiring a probability value, though the values must sum to 1.

Question 3

For a joint PMF, the conditional PMF p(y | x) satisfies which property for each fixed x?

Accepted Answer

∑_y p(y | x) = 1

Answer

For each fixed x, p(y | x) is a valid PMF over y, so it must sum to 1 over all values of y.

Question 4

Given p(1,1)=0.2, p(1,2)=0.3, p(2,1)=0.1, p(2,2)=0.4, what is the conditional PMF p(Y=1 | X=1)?

Accepted Answer

0.4

Answer

p_X(1) = 0.2 + 0.3 = 0.5, so p(Y=1 | X=1) = p(1,1)/p_X(1) = 0.2/0.5 = 0.4.

Question 5

Two discrete random variables X and Y have Cov(X, Y) = 0. This means:

Accepted Answer

X and Y are uncorrelated, but not necessarily independent

Answer

Zero covariance implies uncorrelatedness, but independence is a stronger condition; uncorrelated variables can still be dependent.

PMF - Probability Mass Function Practice Test