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Let $X$ have the Negative Binomial distribution with parameters $r$ and $p$. The Negative Binomial distribution is a mixture distribution or compound distribution. That is $X$ is $\text{Poisson}(\lambda)$ where $\lambda$ is randomly chosen from a $\text{Gamma}(r, p/(1-p))$. Use this relation to write an R function for randomly drawing $n$ Negative Binomial random samples. Hint: Use the R functions rpois and rgamma.

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If this is homework, please tag it accordingly. Someone will probably help you provided you indicate where you get in trouble. – chl♦ Nov 28 '11 at 2:21
If it's indeed a homework Q, you certainly get more fun questions than I do. :) – Roman Luštrik Nov 28 '11 at 9:13
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As indicated in the OP's the comment to Xi'an's post, I have added the homework tag to this question. – cardinal Nov 28 '11 at 13:56

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The answer is contained in the description of the mixture decomposition of the negative binomial distribution as a Poisson distribution where the parameter is itself random with a Gamma distribution. How much more of a hint do you need?

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yes this is hoemwork. I used rgamma to initially generate a random variable. that random variable i allotted to the value of lamdba and used that for the lambda in rpois. But it doesnt work. Pls tell me where im wrong. – probabilityman Nov 28 '11 at 13:53
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x <- rgamma(1, shape=2, rate=0.1) x lambda <- x ylim <- c(0:1) y1 <- rpois(n, lambda) plot (y1, ylab="probability" ,ylim= ylim, type= 'hist'' ) – probabilityman Nov 28 '11 at 13:53
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@cardinal: to me it should be an answer, namely that it is not the purpose of the forum to do someone's homework. Comments are prone to be overlooked. – Xi'an Nov 28 '11 at 14:11
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@probabilityman: your R code simulates one single $\lambda$ and reuses it in $n$ Poisson $\mathcal{P}(\lambda)$ simulations. This is not what is indicated by the compound representation of the negative binomial random variable, compound of one Poisson variable and of one Gamma variable. – Xi'an Nov 28 '11 at 14:13
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I agree we should not be doing other people's homework for them. That was not the intent of my comment. :) I think @whuber's edit clarifies the post a bit. Cheers. – cardinal Nov 28 '11 at 18:32
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If this is not a homework question, you're probably better off with the rnbinom function from stats. If you're lucky (I haven't checked) you can check it's source to see how the implementers did it.

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An efficient negative binomial generator bypasses the Poisson/Gamma representation, using instead optimised accept-reject techniques. – Xi'an Nov 28 '11 at 13:39

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