25  Binomial and Negative Binomial Distributions

25.1 Bernoulli trials

In a sequence of Bernoulli($p$) trials:

• There are only two possible outcomes, “success” (1) and “failure” (0), on each trial.
• The unconditional/marginal probability of success is the same on every trial, and equal to $p$.
• The trials are independent.

25.2 Binomial distributions

Example 25.1 In a random bit stream, the probability that any particular bit is 1 is 0.75, independently of all other bits. Let $X$ be the number of 1s in the next 10 bits.

1. Identify by name the distribution of $X$. Be sure to specify the values of relevant parameters.
   
   
   
   
   

2. Compute and interpret $\text{P}(X=6)$.
   
   
   
   
   

3. Compute and interpret $\text{E}(X)$.
   
   
   
   
   

• A discrete random variable $X$ has a Binomial distribution with parameters $n$, a nonnegative integer, and $p\in [0,1]$ if its probability mass function is $$\begin{array}{rlr}{p}_X(x)& =(\genfrac{}{}{0ex}{}{n}{x})p^x(1-p{)}^{n-x},& x=0,1,2,\dots ,n\end{array}$$ If $X$ has a Binomial($n$, $p$) distribution $$\begin{array}{rl}\text{E}(X)& =np\\ \text{Var}(X)& =np(1-p)\end{array}$$
• If $X$ counts the number of successes in a fixed number, $n$, of Bernoulli($p$) trials then $X$ has a Binomial($n,p$) distribution.

Example 25.2 In each of the following situations determine whether or not $X$ has a Binomial distribution. If so, specify $n$ and $p$. If not, explain why not.

1. Roll a die 20 times; $X$ is the number of times the die lands on an even number.
   
   
   

2. Roll a die 20 times; $X$ is the number of times the die lands on 6.
   
   
   

3. Roll a die until it lands on 6; $X$ is the total number of rolls.
   
   
   

4. Roll a die 20 times; $X$ is the sum of the numbers rolled.
   
   
   

5. Shuffle a standard deck of 52 cards (13 hearts, 39 other cards) and deal 5 without replacement; $X$ is the number of hearts dealt. (Hint: be careful about why.)
   
   
   

6. Roll a fair six-sided die 10 times and a fair four-sided die 10 times; $X$ is the number of 3s rolled (out of 20).
   
   
   

25.3 Geometric distributions

Example 25.3 Maya is a basketball player who makes 40% of her three point field goal attempts. Suppose that she attempts three pointers until she makes one and then stops. Let $X$ be the total number of shots she attempts. Assume shot attempts are independent.

1. Does $X$ have a Binomial distribution? Why or why not?
   
   
   
   
   

2. What are the possible values that $X$ can take? Is $X$ discrete or continuous?
   
   
   
   
   

3. Compute and interpret $\text{P}(X=3)$.
   
   
   
   
   

4. Find the probability mass function of $X$.
   
   
   
   
   

5. Construct a table, plot, and spinner representing the distribution of $X$.
   
   
   
   
   

6. Compute and interpret $\text{P}(X>5)$ without summing. (Hint: consider the first 5 trials.)
   
   
   
   
   

7. What seems like a reasonable general formula for $\text{E}(X)$? Make a guess, and then compute and interpret $\text{E}(X)$ for this example.
   
   
   
   
   

8. Would $\text{Var}(X)$ be bigger or smaller if $p=0.9$? If $p=0.1$?
   
   
   
   
   

9. Compute and interpret $\text{P}(X>15|X>10)$. What do you notice?
   
   
   
   
   

• A discrete random variable $X$ has a Geometric distribution with parameter $p\in [0,1]$ if its probability mass function is $$\begin{array}{rlr}{p}_X(x)& =p(1-p{)}^{x-1},& x=1,2,3,\dots \end{array}$$ If $X$ has a Geometric($p$) distribution $$\begin{array}{rl}\text{E}(X)& =\frac{1}{p}\\ \text{Var}(X)& =\frac{1-p}{p^2}\end{array}$$
• Suppose you perform Bernoulli($p$) trials until a single success occurs and then stop; let $X$ be the total number of trials performed, including the success. Then $X$ has a Geometric($p$) distribution. In this situation, exactly $x$ trials are performed if and only if
  • the first $x-1$ trials are failures, and
  • the $x$th (last) trial results in success.

25.4 Negative Binomial Distributions

Example 25.4 Maya is a basketball player who makes 86% of her free throw attempts. Suppose that she attempts free throws until she makes 5 and then stops. Let $X$ be the total number of free throws she attempts. Assume shot attempts are independent.

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. Compute and interpret $\text{P}(X=5)$
   
   
   
   
   

4. Compute and interpret $\text{P}(X=6)$
   
   
   
   
   

5. Compute and interpret $\text{P}(X=7)$
   
   
   
   
   

6. Compute and intepret $\text{P}(Y=8)$
   
   
   
   
   

7. Find the probability mass function of $X$.
   
   
   
   
   

8. What seems like a reasonable general formula for $\text{E}(X)$? Interpret $\text{E}(X)$ for this example.
   
   
   
   
   

9. Would the variance be larger or smaller if attempted free throws until she made 10 instead of 5?
   
   
   
   
   

• A discrete random variable $X$ has a Negative Binomial distribution with parameters $r$, a positive integer, and $p\in [0,1]$ if its probability mass function is $$\begin{array}{rlr}{p}_X(x)& =(\genfrac{}{}{0ex}{}{x-1}{r-1})p^r(1-p{)}^{x-r},& x=r,r+1,r+2,\dots \end{array}$$ If $X$ has a NegativeBinomial($r$, $p$) distribution $$\begin{array}{rl}\text{E}(X)& =\frac{r}{p}\\ \text{Var}(X)& =\frac{r(1-p)}{p^2}\end{array}$$
• Suppose you perform a sequence of Bernoulli($p$) trials until $r$ successes occur and then stop. Let $X$ be the total number of trials performed, including the trials on which the successes occur. Then $X$ has a NegativeBinomial($r$,$p$) distribution. In this situation, exactly $x$ trials are performed if and only if
  • there are exactly $r-1$ successes among the first $x-1$ trials, and
  • the $x$th (last) trial results in success.
• There are $(\genfrac{}{}{0ex}{}{x-1}{r-1})$ possible sequences that satisfy the above, and each of these sequences — with $r$ successes and $x-r$ failures — has probability $p^r(1-p{)}^{x-r}$.

Example 25.5 What is another name for a NegativeBinomial(1,$p$) distribution?

Example 25.6 Suppose $X_1,\dots ,X_r$ are independent each with a Geometric($p$) distribution. What is the distribution of $X_1+\cdots +X_r$? Find the expected value and variance of this distribution.

Example 25.7 Each box of a certain type of cereal contains one of $n$ distinct prizes — labeled $1,\dots ,n$ — and you want to obtain a complete set. Suppose that each box of cereal is equally likely to contain any one of the $n$ prizes, independently from box to box. You purchase cereal boxes one box at a time until you have the complete set of $n$ prizes. Let $X$ be the number of boxes purchased to obtain a complete set of prizes. Compute and interpret $\text{E}(X)$ as a function of $n$.