PMF Joint and Marginal PMFs 2 — Questions and Answers
Question 1: If X and Y are independent discrete random variables, what is E[XY]?
- E[X] + E[Y]
- E[X] · E[Y] (Correct answer)
- E[X²] · E[Y²]
- E[X] / E[Y]
Correct answer: E[X] · E[Y]
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?
- 2
- 3
- 5
- 6 (Correct answer)
Correct answer: 6
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?
- ∑_y p(y | x) = p_X(x)
- ∑_y p(y | x) = 1 (Correct answer)
- ∑_y p(y | x) = p_X(x) · p_Y(y)
- ∑_x ∑_y p(y | x) = 0
Correct answer: ∑_y p(y | x) = 1
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)?
- 0.2
- 0.5
- 0.4 (Correct answer)
- 0.3
Correct answer: 0.4
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:
- X and Y are always independent
- X and Y are uncorrelated, but not necessarily independent (Correct answer)
- X and Y must have the same distribution
- The joint PMF must factor as a product of marginals
Correct answer: X and Y are uncorrelated, but not necessarily independent
Zero covariance implies uncorrelatedness, but independence is a stronger condition; uncorrelated variables can still be dependent.
Question 6: If p(x, y) = p_X(x) · p_Y(y) for all (x, y), which of the following is also necessarily true?
- Cov(X, Y) = 1
- E[X + Y] = 0
- Var(X + Y) = Var(X) + Var(Y) (Correct answer)
- E[X] = E[Y]
Correct answer: Var(X + Y) = Var(X) + Var(Y)
When X and Y are independent, Cov(X,Y) = 0, so Var(X + Y) = Var(X) + Var(Y).
Question 7: The law of total probability for PMFs states that p_X(x) can be written as:
- p_X(x) = p(x, y) / p_Y(y)
- p_X(x) = ∑_y p(x | y) · p_Y(y) (Correct answer)
- p_X(x) = ∏_y p(x | y)
- p_X(x) = p(x, y) · p_Y(y)
Correct answer: p_X(x) = ∑_y p(x | y) · p_Y(y)
By the law of total probability, p_X(x) = ∑_y p(x | y) · p_Y(y), which expands the marginal using conditional probabilities.
PMF Joint and Marginal PMFs 2
If X and Y are independent discrete random variables, what is E[XY]?