Negative binomial distribution

Given independent Bernoulli trials with probability of success $ p$, the frequency function of the number of trials until the $ r$-th success is

$\displaystyle p(k) = \binom{k-1}{r-1} p^r (1-p)^{k-r},
\quad k = r,r+1,\ldots.
$

This is called a negative binomial distribution with parameter $ (r,p)$.

The frequency function and the cumulative distribution function (CDF) with parameter $ r =$ and $ p =$ , are displayed up to the value of outcome.

The geometric distribution is a special case of negative binomial distribution when $ r = 1$. Moreover, if $ X_1,\ldots,X_r$ are independent and identically distributed (iid) geometric random variables with parameter $ p$, then the sum

$\displaystyle Y = \sum_{i=1}^r X_i$ (3)

becomes a negative binomial random variable with parameter $ (r,p)$.

The frequency function and the cumulative distribution function can be shown graphically.

Expectation, variance and mgf of negative binomial distribution. By using the sum of iid geometric rv's we can compute the expectation, the variance, and the mgf of negative binomial random variable $ Y$.

$\displaystyle E[Y]$ $\displaystyle = E\left[\sum_{i=1}^r X_i\right] = \sum_{i=1}^r E[X_i] = \frac{r}{p};$    
$\displaystyle {\rm Var}(Y)$ $\displaystyle = {\rm Var}\left(\sum_{i=1}^r X_i\right) = \sum_{i=1}^r {\rm Var}(X_i) = \frac{r(1-p)}{p^2};$    
$\displaystyle M_Y(t)$ $\displaystyle = M_{X_1}(t)\times M_{X_2}(t)\times\cdots\times M_{X_n}(t) = \left[\frac{pe^t}{1 - (1 - p)e^t}\right]^r .$    

Example 4. What is the average number of times one must throw a die until the outcome ``1'' has occurred 4 times?



  Generated by MATH GO: 2005-11-16