18  Negative Binomial Distributions

Example 18.1 Suppose that 86% of Cal Poly students are CA residents. Randomly select Cal Poly students one at a time, independently¹, until you select 5 CA residents, then stop. Let \(X\) be the total number of students selected.

1. Does \(X\) have a Binomial distribution? A Geometric distribution? Why or why not?
   
   
   
   
   
2. What are the possible values of \(X\)? Is \(X\) discrete or continuous?
   
   
   
   
   
3. Describe in words how you could use simulation to approximate the distribution of \(X\).
   
   
   
   
   
4. Compute and interpret \(\text{P}(X=5)\).
   
   
   
   
   
5. Compute the probability that the first student is not a CA resident but the next 5 are.
   
   
   
   
   
6. Why is \(\text{P}(X=6)\) different from the probability in the previous part?
   
   
   
   
   
7. How many outcomes satisfy the event \(\{X=6\}\)? Answer without listing all the possibilities. Hint: careful, it’s not \(\binom{6}{5}\); why not?
   
   
   
   
   
8. Compute and interpret \(\text{P}(X=6)\)
   
   
   
   
   
9. Compute the probability that the first two students are not CA residents but the next 5 are.
   
   
   
   
   
10. Why is \(\text{P}(X=7)\) different from the probability in the previous part?
    
    
    
    
    
11. How many outcomes satisfy the event \(\{X=7\}\)? Answer without listing all the possibilities. Hint: careful, it’s not \(\binom{7}{5}\); why not?
    
    
    
    
    
12. Compute and interpret \(\text{P}(X=7)\)
    
    
    
    
    
13. Suggest a formula for the probability mass function of \(X\). Then use the pmf to create a table representing the distribution of \(X\).
    
    
    
    
    
14. What seems like a reasonable shortcut formula for \(\text{E}(X)\)?
    
    
    
    
    
15. Use the theoretical pmf to compute \(\text{E}(X)\). Did the shortcut formula work?
    
    
    
    
    
16. Interpret \(\text{E}(X)\) for this example.
    
    
    
    
    
17. Compute \(\text{Var}(X)\).
    
    
    
    
    
18. Would the variance be larger or smaller if we kept selecting until we obtain 10 CA residents instead of 5?
    
    
    
    
    

• Perform Bernoulli(\(p\)) trials until a fixed number, \(r\), of successes and then stop. Let \(X\) count the total number of trials, including the \(r\) successes. The distribution of \(X\) is defined to be the Negative Binomial(\(r\), \(p\)) distribution.
• A Negative Binomial distribution is specified by two parameters:
  • \(r\) (an integer): the fixed number of successes
  • \(p\) (in \([0, 1]\)): the fixed marginal probability of success on each Bernoulli trial, or the population proportion of success
• A discrete random variable \(X\) has a Negative Binomial distribution with parameters \(r\), a positive integer, and \(p\in[0, 1]\) if and only if its probability mass function is \[\begin{align*} p_{X}(x) & = \binom{x-1}{r-1}p^r(1-p)^{x-r}, & x=r, r+1, r+2, \ldots \end{align*}\]
  • In the NegativeBinomial(\(r\), \(p\)) situation, there are \(\binom{x-1}{x-r}=\binom{x-1}{r-1}\) S/F sequences that result in exactly \(x-r\) failures—and thus \(r-1\) succeses—in the first \(x-1\) trials and success on trial \(x\)
  • and each of these sequences has probability \(p^r(1-p)^{x-r}\)
• If \(X\) has a NegativeBinomial(\(r\), \(p\)) distribution \[\begin{align*} \text{E}(X) & = \frac{r}{p}\\ \text{Var}(X) & = \frac{r(1-p)}{p^2} \end{align*}\]

Example 18.2 What is another name for a NegativeBinomial(1, \(p\)) distribution?

Example 18.3 Xavier bets on red on roulette until he wins 3 bets and then he leaves. After Xavier leaves, Zander arrives and bets on black until he wins 2 bets and then he leaves. Let \(X\) be the number of bets that Xavier makes and let \(Z\) be the number of bets that Zander makes.

1. Identify the distribution of \(X\) and the distribution of \(Z\).
   
   
   
   
   
2. Are \(X\) and \(Z\) independent? Explain.
   
   
   
   
   
3. What does \(X+Z\) represent in this context? Identify the distribution of \(X+Z\) without doing any calculations.
   
   
   
   
   
4. Suppose that Yolanda starts making bets on the same game as Xavier, and she bets on black until she wins 5 bets. Does \(X + Y\) have a Negative Binomial distribution? Explain.
   
   
   
   
   
5. Suppose instead that Zavier bets on the number 7 until he wins 5 bets. Does \(X + Z\) have a Negative Binomial distribution in this case? Explain.
   
   
   
   
   

• If \(X\) and \(Y\) are independent, \(X\) has a NegativeBinomial(\(r_X\), \(p\)) distribution, and \(Y\) has a NegativeBinomial(\(r_Y\), \(p\)) distribution then \(X+Y\) has a NegativeBinomial(\(r_X+r_Y\), \(p\))
• A NegativeBinomial(1, \(p\)) distribution is a Geometric(\(p\)) distribution.
• If \(X_1, X_2, \ldots, X_r\) are independent each with a Geometric(\(p\)) distribution, then \(X_1+\cdots+X_r\) has a NegativeBinomial(\(r, p\)) distribution.

In a Negative Binomial situation, the number of successes is fixed so what’s random is the number of failures and hence the number of trials. Unfortunately, there is not a standard definition of Negative Binomial in terms of what is counted: failures or trials. It’s a good idea to always check the documentation of a software package to see what definition is used.

• Perform Bernoulli(\(p\)) trials until a fixed number, \(r\), of successes and then stop. Let \(X\) count the total number of failures (excluding the \(r\) successes). The distribution of \(X\) is defined to be the Pascal(\(r\), \(p\)) distribution.
  • The possible values of a NegativeBinomial(\(r\), \(p\)) distribution are \(r, r+1, r+2,\ldots\)
  • The possible values of a Pascal(\(r\), \(p\)) distribution are \(0, 1, 2,\ldots\)
• If \(X\) has a NegativeBinomial(\(r\), \(p\)) distribution then \((X-r)\) has a Pascal(\(r\), \(p\)) distribution.
• Similarly, if \(Y\) has a Pascal(\(r\), \(p\)) distribution then \((Y+r)\) has a NegativeBinomial(\(r\), \(p\)) distribution.
• Some software packages call “Negative Binomial” what we are calling “Pascal”.

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1. Technically this means with replacement, but since the population of Cal Poly students is much larger than the sample we’re selecting then independence is a reasonable assumption even if selecting without replacement.