Negative Binomial Distribution

Question 0

What is the variance of a negative binomial distribution?

Answer 0

r(1-p)/p^2

Where r is the number of successes and p is the probability of success.

Question 1

What is the expected value of a negative binomial distribution?

Answer 1

r/p

where r is the number of successes and p is the probability of success.

Question 2

What is the PMF of the negative binomial random variable in terms of non-interesting events (Y)?

Answer 2

P(Y=y) = ((r+y-1)C(y)) p^r (1-p)^y

Where y is the number of non-interesting events, p is the probability of success and r is the number of successes.

Question 3

What is the PMF of the negative binomial distribution with respect to number of trials?

Answer 3

P(X=n) = ((n-1)C(r-1)) p^r (1-p)^n-r

Where p is the probability of success, n is the number if trials and r is the number of successes.

Question 4

Repeatedly roll a fair die until the outcome 3 has occurred for the 4th time. Let X be the number of times needed in order to achieve the goal. Find E(x) and Var(x).

Answer 4

E(x) = 24
Var(x) = 119.79

Question 5

Repeatedly roll a fair die until the outcome 3 has occurred for the 4th time. Let X be the number of times needed in order to achieve the goal. Find E(x) and Var(x).

Answer 5

What type of distribution is needed to solve this problem?

Question 6

Consider a 52-card deck. Repeatedly draw a card with replacement and record it’s face value. Let X be the number of trials needed to get 3 kings.

What type of distribution is needed to solve this question and what is the PMF?

Answer 6

Negative Binomial

P(X=n) = ((n-1)C(r-1)) p^r (1-p)^(n-r)

Question 7

Consider a 52-card deck. Repeatedly draw a card with replacement and record it’s face value. Let X be the number of trials needed to get 3 kings.

What is the probability that X=39?

Answer 7

.0179

Question 8

What is the moment generating function of the negative binomial distribution?

Answer 8

(p/(1-qe^t))^r