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

E[X] + E[Y]

Answer

E[X²] · E[Y²]

Answer

E[X] / E[Y]

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

2

Answer

3

Answer

5

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

∑_y p(y | x) = p_X(x)

Answer

∑_y p(y | x) = p_X(x) · p_Y(y)

Answer

∑_x ∑_y p(y | x) = 0

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

0.2

Answer

0.5

Answer

0.3

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

X and Y are always independent

Answer

X and Y must have the same distribution

Answer

The joint PMF must factor as a product of marginals

Question 6

If p(x, y) = p_X(x) · p_Y(y) for all (x, y), which of the following is also necessarily true?

Accepted Answer

Var(X + Y) = Var(X) + Var(Y)

Answer

Cov(X, Y) = 1

Answer

E[X + Y] = 0

Answer

E[X] = E[Y]

Question 7

The law of total probability for PMFs states that p_X(x) can be written as:

Accepted Answer

p_X(x) = ∑_y p(x | y) · p_Y(y)

Answer

p_X(x) = p(x, y) / p_Y(y)

Answer

p_X(x) = ∏_y p(x | y)

Answer

p_X(x) = p(x, y) · p_Y(y)