PMF PMF Properties and Axioms 2

Question 1

How is the CDF F(x) related to the PMF P(X=k) for discrete random variables?

Accepted Answer

F(x) = Σ P(X=k) for all k ≤ x

Answer

The CDF is the cumulative sum of the PMF: F(x) = P(X ≤ x) = Σ P(X=k) for all k ≤ x.

Question 2

For a PMF with values {1,2,3,4}, if F(3) = 0.8 and F(4) = 1, what is P(X=4)?

Accepted Answer

0.2

Answer

P(X=4) = F(4) - F(3) = 1.0 - 0.8 = 0.2.

Question 3

What property of the CDF makes it easy to compute P(a < X ≤ b)?

Accepted Answer

P(a < X ≤ b) = F(b) - F(a)

Answer

P(a < X ≤ b) = F(b) - F(a) is a direct consequence of CDF being a cumulative sum.

Question 4

A PMF assigns P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3. What is F(2)?

Accepted Answer

0.7

Answer

F(2) = P(X ≤ 2) = P(X=1) + P(X=2) = 0.2 + 0.5 = 0.7.

Question 5

For a discrete PMF, what type of function is the CDF?

Accepted Answer

A right-continuous step function

Answer

The CDF of a discrete distribution is a step function that jumps at each mass point and is right-continuous.

PMF - Probability Mass Function Practice Test