PMF Negative Binomial Distribution 2

Question 1

For a Negative Binomial with r=2 and p=0.5, what is E[X] (failures before 2nd success)?

Accepted Answer

2

Answer

E[X] = r(1-p)/p = 2 × 0.5/0.5 = 2 failures expected before the 2nd success.

Question 2

How does the Negative Binomial PMF relate to the Binomial PMF?

Accepted Answer

They count different things: Binomial counts successes in fixed trials; Negative Binomial counts trials to fixed successes

Answer

The Binomial fixes n trials and counts k successes, while the Negative Binomial fixes r successes and counts the number of trials needed.

Question 3

For the Negative Binomial (total trials until r-th success), what is the PMF?

Accepted Answer

P(X=k) = C(k-1, r-1) × p^r × (1-p)^(k-r)

Answer

P(X=k) = C(k-1, r-1) × p^r × (1-p)^(k-r) for total trials needed, where the last trial is always the r-th success.

Question 4

A researcher needs 5 volunteer sign-ups. Each person contacted has a 30% chance of volunteering. What is E[total people contacted]?

Accepted Answer

5/0.3 ≈ 16.67

Answer

For total trials until r successes, E[X] = r/p = 5/0.3 ≈ 16.67 people need to be contacted.

Question 5

What is the support of the Negative Binomial (total trials until r-th success)?

Accepted Answer

{r, r+1, r+2, …}

Answer

The minimum number of trials to get r successes is r (all trials are successes), so the support starts at k = r.

PMF - Probability Mass Function Practice Test