- Probability mass function
- Moment Generating Function
- Mean of Negative Binomial
- Variance of Negative Binomial Distribuiton
Consider the situation where one performs a number Bernoulli trials, each trial has a probability of success , and
trials continue until the th successs occurs. Let be the random variable which is the number of trials up to
and including the th success. This means that the range of X is the set . Then the pmf would be given by
Note that there are r successes and x-r previous failures, with the last success at a fixed rth position. Thus the number of possible outcomes is the number of combinations of selecting objects taken at a time.
What is the mgf of the negative binomial distribution? Let us compute the value of
Writing the first few terms of the expansion, we get
which simplifies to
and which upon factoring out and further simplication results in
It will be shown later that the bracketed terms is equivalent to
, see the blog entry A negative binomial series identity and therefore the moment generating function of the negative binomial distribution
is given by
Note that the numerator in the formula of Spiegel's Statistics in page. 118 should be raised to the power .
The expectation or mean of the negative binomial distribution with the pmf and mgf above is obtained by differentiating the mgf wrt t and setting t to zero:
When differentiated, the derivative is
and the value at t=0 is
which collapses to
The second moment or of the negative binomial distribution with the pmf and mgf above is obtained by differentiating the mgf twice wrt t and setting t to zero and the variance is computed as
We will leave as an exercise (at the moment since it is so tedious) that the variance is given by
This entry is subject to review but the final formulas are all right.
Feb. 20, 2010: We missed the square in the denominator! We will redo the presentation for the variance.