2.8.1 Negative Binomial

Question 1

what does the negative binomial distribution measure?

Answer 1

The negative binomial distribution is a discrete distribution of the number of independent Bernoulli trials it takes to get a specified number of “successes’’ to occur.

Negative Binomial(r,p) where r is the desired number of successes and p is the probability

Question 2

which distribution does the negative binomial generalize?

Answer 2

the negative binomial distribution generalizes the geometric distribution; from counting “trials” until the first “success”, we generalize to the r-th “success”. Stated differently, when r=1, the negative binomial distribution becomes a geometric distribution.

Question 3

what is the relationship between the negative binomial and geometric distributions?

Answer 3

the negative binomial is the sum of r independent geometric distributions

Question 4

What is the PMF of the negative binomial distribution (trials version)?

Answer 4

p(x) = c(x-1,r-1) * p^r * (1-p)^x-r

Question 5

What does the argument represent in the trials version?

Answer 5

The trial number on which the r-th success occurs.

Question 6

Why is ( \binom{x-1}{r-1} ) used in the PMF?

Answer 6

It counts the ways to get ( r-1 ) successes in the first ( x-1 ) trials, before the final success at trial ( x ).

Question 7

What are the mean and variance of a negative binomial random variable (trials version)?

Answer 7

E(X) = r/p
Var(X) = r((1-p) / (p^2))

Question 8

What is the alternate version of the negative binomial?

Answer 8

It count failures instead of trails.

p(y) = c(y+r-1,y) * p^r * (1-p)^y

Question 9

What are the mean and variance for the failure-count version?

Answer 9

E(Y) = r(1/p - 1)
Var(Y) = r((1-p) / p^2)