3.5.2 The Negative Binomial Distribution

Conditions for Negative Binomial Random Variable and Distribution:

1. Independent Trials:
   • The experiment consists of a sequence of independent trials.
2. Binary Outcomes:
   • Each trial can result in either a success (S) or a failure (F).
3. Constant Probability:
   • The probability of success is constant across trials, denoted as $P(S\text{ on trial }i)=p$ for $i=1,2,3,\dots$.
4. Fixed Number of Successes:
   • The experiment continues until a total of $r$ successes have been observed, where $r$ is a specified positive integer.

• Random variable of interest:

  • $X$ = number of failures that precede the $r$th success
  • Called a negative binomial random variable
  • Contrast with binomial random variable:
    • Number of successes is fixed
    • Number of trials is random

• Possible values of $X$:

  • $0,1,2,\dots$

• Notation:

  • $nb(x;r;p)$: pmf of $X$

• Example:

  • $nb(7;3;p)$ = $P(X=7)$
    • Probability that exactly 7 failures occur before the 3rd success
  • Conditions:
    • 10th trial must be a success
    • Exactly 2 successes among the first 9 trials
  • Calculation:

$$\begin{array}{rl}nb(7;3;p)& =\left\{\left(\begin{array}{l}9\\ 2\end{array}\right)\cdot p^2(1-p{)}^7\right\}\cdot p\\ & =\left(\begin{array}{l}9\\ 2\end{array}\right)\cdot p^3(1-p{)}^7\end{array}$$

• Generalization:
  • Formula for the negative binomial pmf

Proposition

The pmf of the negative binomial rv $X$ with parameters $r=$ number of $S$ ‘s and $p=P\left(S\right)$ is, for all $x=0,1,2,\dots$, $nb\left(x;r,p\right)=\left(\begin{array}{c}x+r-1\\ r-1\end{array}\right)p^r{\left(1-p\right)}^x$

EX 3.37

• Alternative definition:

  • Negative binomial random variable may be defined as:
  • Number of trials $X+r$ rather than number of failures $X$

• Special case: $r=1$

  • pmf:

  {nb}(x; 1, p) = (1 - p)^{x} p, \quad x = 0, 1, 2, \ldots \tag{3.17}

• Relation to previous EX 3.12 gender of newborn child:

  • Derived pmf for number of trials to obtain the first success
  • Similar to Expression (3.17)
  • Both:
    • $X$ = number of failures
    • $Y$ = number of trials $(=1+X)$
    • Referred to as geometric random variables
  • pmf in Expression (3.17):
    • Called the geometric distribution

• Expected values:

  • Expected number of trials until the first success:
    • Shown in Example 3.19 to be $1/p$
  • Expected number of failures until the first success:
    • $\frac{1}{p}-1=\frac{1-p}{p}$
  • Intuitive expectation:
    • $r\cdot \frac{1-p}{p}$ failures before the $r$th success
    • This is $E(X)$

• Variance:

  • Simple formula for $V(X)$

Proposition

If $X$ is a negative binomial rv with $\mathrm{pmf}nb\left(x;r,p\right)$ , then

• $E\left(X\right)=\frac{r\left(1-p\right)}{p}$
• $V\left(X\right)=\frac{r\left(1-p\right)}{p^2}$

Finally, by expanding the binomial coefficient in front of $p^r{\left(1-p\right)}^x$ and doing some cancellation, it can be seen that $nb\left(x;r;p\right)$ is well defined even when $r$ is not an integer.

This generalized negative binomial distribution has been found to fit observed data quite well in a wide variety of applications.