## What is the MGF of negative binomial distribution?

The something is just the mgf of the geometric distribution with parameter p. So the sum of n independent geometric random variables with the same p gives the negative binomial with parameters p and n. for all nonzero t. Another moment generating function that is used is E[eitX].

**What is the MGF of a binomial distribution?**

The Moment Generating Function of the Binomial Distribution (3) dMx(t) dt = n(q + pet)n−1pet = npet(q + pet)n−1. Evaluating this at t = 0 gives (4) E(x) = np(q + p)n−1 = np.

### Can a moment generating function be negative?

It does not appear to be widely known that the moment-generating function Mx (t) contains information about negative as well as positive integer moments.

**What is the pdf of a binomial distribution?**

A representative example of a binomial probability density function (pdf) is plotted below for a case with p=0.3 and N=12, and provides the probability of observing -1, 0, 1, …, 11, or 12 heads. Note, as expected, there is 0 probability of obtaining fewer than 0 heads or more than 12 heads.

## What is the mean and variance of negative binomial distribution?

The mean of the negative binomial distribution with parameters r and p is rq / p, where q = 1 – p. The variance is rq / p2. The simplest motivation for the negative binomial is the case of successive random trials, each having a constant probability P of success.

**What is a negative binomial random variable?**

A negative binomial random variable is the number X of repeated trials to produce r successes in a negative binomial experiment. The probability distribution of a negative binomial random variable is called a negative binomial distribution. Suppose we flip a coin repeatedly and count the number of heads (successes).

### How to derive the MGF of a negative binomial distribution?

The probability mass function for the geometric distribution is [math]q^ {x-1}*p [/math] where [math]q = 1- p [/math] and this will be used in deriving the MGF for the geometric distribution which will then later be used to derive the MGF for the negative binomial distribution.

**Is the negative binomial distribution infinitely divisible?**

The negative binomial distribution is infinitely divisible, i.e., if Y has a negative binomial distribution, then for any positive integer n, there exist independent identically distributed random variables Y 1., Y n whose sum has the same distribution that Y has.

## When is a negative binomial distribution a Poisson mixture?

Gamma–Poisson mixture. The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution.

**What is the negative binomial distribution in Bernoulli?**

Waiting time in a Bernoulli process. For the special case where r is an integer, the negative binomial distribution is known as the Pascal distribution. It is the probability distribution of a certain number of failures and successes in a series of independent and identically distributed Bernoulli trials.