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

∑_x p(x, y) = 1 for each fixed y

Answer

p(x, y) = p(x) · p(y) for all x, y

Answer

∑_x ∑_y p(x, y) = 0

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

Multiplying p(x, y) by p(y) for each y

Answer

Dividing p(x, y) by p(y) for each y

Answer

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

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

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

Answer

p(x | y) = p(y | x) for all x, y

Answer

E[XY] = 0

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

1/6

Answer

1/4

Answer

1/3

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

0.1

Answer

0.4

Answer

0.7

Question 6

For the joint PMF p(x, y) = 1/9 · x · y with x, y ∈ {1, 2}, are X and Y independent?

Accepted Answer

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

Answer

No, because E[XY] ≠ E[X]E[Y]

Answer

No, because the marginals are not uniform

Answer

Yes, because the joint PMF is symmetric

Question 7

The conditional PMF of X given Y = y is defined as:

Accepted Answer

p(x | y) = p(x, y) / p_Y(y), provided p_Y(y) > 0

Answer

p(x | y) = p(x, y) · p_Y(y)

Answer

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

Answer

p(x | y) = p_X(x) · p(x, y)