Home page for accesible maths 5 Models for discrete random variables 5.6.1 Poisson approximation to the Binomial 5.8 Hypergeometric random variables (not examinable)

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5.7 Negative Binomial random variables (not examinable)

Consider an experiment for which the random variable of interest, ${R}$ , corresponds to the number of fails to occur before the ${k}$ th success is obtained in a series of independent Bernoulli trials, each with probability of success being ${\theta}$ .

Here the sample space when ${k=2}$ is ${\{SS,FSS,SFS,FFSS,FSFS,SFFS,\ldots\}}$ , and the induced sample space is ${\{0,1,2,\ldots\}}$ . This is a slightly more general but similar to the geometric random variable case. It might appear the same as Binomial random variable, but the subtle difference is that the last trial must be a success.

For ${r=0,1,2,\dots}$

$\displaystyle p(r)$	$\displaystyle=$	$\displaystyle{\rm P}(k-1\mbox{ successes in first }k+r-1\mbox{ trials, and % success on last})$
	$\displaystyle=$	$\displaystyle{\rm P}(k-1\mbox{ successes in first }k+r-1\mbox{ trials})\times{% \rm P}(\mbox{ success on last})$
	$\displaystyle=$	$\displaystyle\binom{k+r-1}{k-1}(1-\theta)^{r}\theta^{k-1}\times\theta$
	$\displaystyle=$	$\displaystyle\binom{k+r-1}{k-1}(1-\theta)^{r}\theta^{k}.$

It is possible to show that

	$\displaystyle{\rm E}[R]$	$\displaystyle=$	$\displaystyle\frac{k(1-\theta)}{\theta}$
	$\displaystyle{\rm Var}(R)$	$\displaystyle=$	$\displaystyle\frac{k(1-\theta)}{\theta^{2}}.$

Note by setting ${k=1}$ these results are identical to the geometric pmf.