Negative Hypergeometric Distribution
A random variable Yhas a negative hypergeometric distribution if
•Objects are selected without replacement from a population of size N.
•Each member of the population belongs to one of two groups; success or failure. The number
of successes in the population is denoted as rand therefore the number of failures in the
population is N−r.
•Objects are selected from the population until the kth success occurs.
•The random variable Yis defined as the number of the trial on which the kth success occurs.
•The parameters for the hypergeometric random variable Yare value of k, the population size
Nand the number of successes in the population r.
•The probability distribution function of Y is
P(Y=y) = p(y) = y−1
k−1N−y
r−k
N
ry=k, k + 1, . . . , N −r+k
•The theoretical mean of the negative hypergeometric random variable Yis
µ=E(Y) = kN+ 1
r+ 1
•The theoretical variance of the negative hypergeometric random variable Yis
σ2=V(Y) = k(N+ 1)(N−r)(r+ 1 −k)
(r+ 1)2(r+ 2)
Working with negative hypergeometric random variables in R.
R does not include the negative hypergeometric random variable as a built-in function. In order to
work with this random variable, you will need to copy and paste the following function into R.
dneghyper<- function(y,r,N,k){choose(y-1,k-1)*choose(N-y,r-k)/choose(N,r)}
To find a probability P(Y=y) = p(y) for a single value y, the command in R is
dneghyper(y,r,N,k)
To find the probability P(Y≤y), use the sum command to add up all p(y) values for ybetween
and including kand y.
sum(dneghyper(k:y,r,N,k))
To find the probability P(y1≤Y≤y2), use the sum command to add up all p(y) values for y
between and including y1and y2.
sum(dneghyper(y1:y2,r,N,k))
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