What is the moment generating function for the negative binomial distribution?

What is the moment generating function for the negative binomial distribution?

• A. Mx(t) = ((1 - qe^t)/p)^-r
• B. Mx(t) = (pe^t/(1 - q))^r
• C. Mx(t) = (1 - p)^r
• D. Mx(t) = (pe^t)^r

Answer: (A)

The moment generating function (MGF) for the negative binomial distribution is derived based on the properties of the distribution, which counts the number of trials up to and including the r-th success. The negative binomial distribution is characterized by two parameters: the number of successful trials \( r \) and the probability of success on each trial \( p \), where \( q = 1 - p \). The correct moment generating function for a negative binomial distribution is given by: \[ M_X(t) = \left( \frac{1 - q e^t}{p} \right)^{-r} \] This formulation arises from the fact that the MGF is constructed by taking the expected value of the exponential function of the random variable. In this case, by summing the contributions from all the independent Bernoulli trials that compose the negative binomial process, one can arrive at this MGF form. The expression captures the essential components of the distribution: the probability of failure \( q e^t \) reduces the overall probability of generating successes based on the nature of the negative binomial trials. The parameters \( r \) and \( p \) shape how many successes are needed and how likely a success is, respectively

The moment generating function (MGF) for the negative binomial distribution is derived based on the properties of the distribution, which counts the number of trials up to and including the r-th success. The negative binomial distribution is characterized by two parameters: the number of successful trials ( r ) and the probability of success on each trial ( p ), where ( q = 1 - p ).

The correct moment generating function for a negative binomial distribution is given by:

[

M_X(t) = \left( \frac{1 - q e^t}{p} \right)^{-r}

]

This formulation arises from the fact that the MGF is constructed by taking the expected value of the exponential function of the random variable. In this case, by summing the contributions from all the independent Bernoulli trials that compose the negative binomial process, one can arrive at this MGF form.

The expression captures the essential components of the distribution: the probability of failure ( q e^t ) reduces the overall probability of generating successes based on the nature of the negative binomial trials. The parameters ( r ) and ( p ) shape how many successes are needed and how likely a success is, respectively