91Ó°ÊÓ

The generating function is a powerful tool in probability theory and combinatorics. It provides a way to encapsulate an entire sequence of numbers into a single algebraic expression. For the negative binomial distribution, its generating function is crucial to understanding its properties. We start with the probability mass function: \[ f(k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}, \] for \( k = r, r+1, \ldots \). The generating function, denoted by \( G(t) \), is given by the infinite sum: \[ G(t) = \sum_{k=r}^{\infty} f(k) t^k. \] By substituting the mass function into this sum, the generating function becomes: \[ G(t) = \sum_{k=r}^{\infty} \binom{k-1}{r-1} p^r (1-p)^{k-r} t^k. \] To simplify, we factor out constants: \[ G(t) = p^r t^r \sum_{k=r}^{\infty} \binom{k-1}{r-1}((1-p)t)^{k-r}. \] Introducing \( m = k-r \) allows us to use a known series identity, giving the final generating function form:

The mean and variance of a probability distribution are fundamental measures that describe its central tendency and spread. With the generating function \( G(t) = \left(\frac{pt}{1-(1-p)t}\right)^r \), we can derive these statistics for the negative binomial distribution. The mean (expected number of trials needed to achieve \( r \) successes) is calculated as: \[ \text{Mean} = \frac{r(1-p)}{p}. \] This result tells us that the higher the probability of failure \((1-p)\), or the more successes \( r \) desired, the longer it will take. Variance measures how much the number of trials can vary. For this distribution, the variance is: \[ \text{Variance} = \frac{r(1-p)}{p^2}. \] The variance is larger than the mean, indicating that the distribution is more spread out, particularly when the probability \( p \) of success in each trial is low.

The probability mass function (PMF) is key to understanding how probabilities are assigned across possible outcomes in a discrete distribution. For the negative binomial distribution, the PMF is given by: \[ f(k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}, \] where \( k = r, r+1, \ldots \), \( p \) is the probability of success in each trial, and \( r \) is the number of successes you want. The term \( \binom{k-1}{r-1} \) accounts for the different combinations of how the \( r \) successes can be achieved within the trials. The remaining terms \( p^r (1-p)^{k-r} \) provide the probability of those specific successes and failures. Together, these elements reflect the likelihood of requiring exactly \( k \) trials to achieve \( r \) successes when trials are identically and independently carried out.

A series identity is a mathematical expression that allows complex series to be expressed in a simpler form. In this case, it plays a pivotal role in simplifying the generating function for the negative binomial distribution. We use the identity: \[ \sum_{m=0}^{\infty} \binom{m+r-1}{r-1} x^m = \frac{1}{(1-x)^r}, \] by substituting \( x = (1-p)t \). This lets us rewrite part of our sum from the generating function development process. Essentially, it provides a way to switch from a complex summation to a simple fraction, making calculations more manageable. This identity is central to the step where from the complex sum \( \sum_{m=0}^{\infty} \binom{m+r-1}{r-1} ((1-p)t)^m \), we can use the series identity to neatly arrive at the expression: \[ \frac{1}{(1-(1-p)t)^r}. \] It's a crucial step that makes further mathematical deductions, such as those for mean and variance, much easier.