Negative Binomial Distribution
A discrete random variable Yis said to have a negative binomial distribution if
•There are independent and identical trials. Each trial can be thought of as a draw from a
population, where the draw is done with replacement.
•Each trial has two possible outcomes, success and failure.
•The probability of success on each trial is the same, p. Therefore, the probability of failure
on each trial is 1 −p=q.
•The experiment is repeated until the rth success occurs.
•The random variable Y is defined as the number of the trial on which the rth success occurs.
•The parameters for the negative binomial random variable Yare the probability of success
on each trial pand the number of the rth success.
•The probability distribution function of the negative binomial random variable Y is
p(y) = y−1
r−1!pr(1 −p)y−rfor y=r, r + 1, r + 2,...
•The theoretical mean of the negative binomial random variable Y is
µ=E(Y) = r
p
•The theoretical variance of the negative binomial random variable Y is
σ2=V(Y) = r(1 −p)
p2
Working with negative binomial random variables in R.
The built-in function in R for the negative binomial distribution is different from the distribution
described in your textbook. To eliminate any confusion, I have written a function in R called
dnbin that matches the distribution in your textbook. Before you calculate any probabilities for
the negative binomial distribution, you will need to type in this function in R.
dnbin<- function(y,r,p){choose(y-1,r-1)*p^r*(1-p)^(y-r)}
To find a probability P(Y=y) = p(y) for a single value y, the command in R is
dnbin(y,r,p)
To find the probability P(Y≤y), use the sum command to add up all p(y) values for ybetween
and including r and y.
sum(dnbin(r:y,r,p))
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