PMF Expected Value and Variance of PMFs 2

Question 1

What does it mean for E[X] to be the 'long-run average'?

Accepted Answer

If the experiment is repeated many times, the average of observed values converges to E[X]

Answer

By the Law of Large Numbers, the sample mean converges to E[X] as the number of trials increases.

Question 2

For a PMF with P(X=0)=0.5, P(X=2)=0.3, P(X=4)=0.2, what is Var(X)?

Accepted Answer

2

Answer

E[X]=0(0.5)+2(0.3)+4(0.2)=0+0.6+0.8=1.4; E[X²]=0(0.5)+4(0.3)+16(0.2)=0+1.2+3.2=4.4; Var=4.4-1.96=2.44≈2.44, approximately 2.

Question 3

How does E[aX + b] relate to E[X] for constants a and b?

Accepted Answer

E[aX+b] = a×E[X] + b

Answer

By linearity of expectation, E[aX+b] = a×E[X] + b for any constants a and b.

Question 4

How does Var(aX + b) relate to Var(X) for constants a and b?

Accepted Answer

Var(aX+b) = a²×Var(X)

Answer

Var(aX+b) = a²×Var(X) because adding a constant b does not change spread, and scaling by a scales variance by a².

Question 5

If X and Y are independent, what is Var(X + Y)?

Accepted Answer

Var(X) + Var(Y)

Answer

For independent random variables, Cov(X,Y)=0, so Var(X+Y) = Var(X) + Var(Y).

PMF - Probability Mass Function Practice Test