PMF Expected Value and Variance of PMFs

Question 1

How is the expected value E[X] of a discrete random variable X computed from its PMF?

Accepted Answer

E[X] = Σ x × P(X=x) over all x

Answer

E[X] = Σ x × P(X=x) is the probability-weighted sum of all possible values.

Question 2

For a fair six-sided die, what is E[X]?

Accepted Answer

3.5

Answer

E[X] = (1+2+3+4+5+6)/6 = 21/6 = 3.5 for a fair die.

Question 3

How is Var(X) computed from a PMF?

Accepted Answer

Var(X) = E[X²] - (E[X])²

Answer

Var(X) = E[X²] - (E[X])² is the computational formula for variance, using the second moment minus the squared mean.

Question 4

If E[X] = 4 and E[X²] = 20, what is Var(X)?

Accepted Answer

4

Answer

Var(X) = E[X²] - (E[X])² = 20 - 16 = 4.

Question 5

What is the standard deviation of X if Var(X) = 9?

Accepted Answer

3

Answer

SD(X) = √Var(X) = √9 = 3.

PMF - Probability Mass Function Practice Test