10 Binomial Distributions

Some probabilistic situations are so common that they have special names.
In a sequence of Bernoulli($p$) trials
- There are only two possible outcomes, “success” (1) or not (0, “failure”), on each trial.
- The unconditional/marginal probability of success is the same on every trial, and equal to $p$.
- The trials are independent.
There are several commonly used distributions associated with Bernoulli trials, including Geometric and Binomial.

Example 10.1 Consider an extremely simplified model for the daily closing price of a certain stock. Every day the price either goes up or goes down, and the movements are independent from day-to-day. Assume that the probability that the stock price goes up on any single day is 0.55. Let $X$ be the number of days on which the price goes up in a 5-day week.

Compute and interpret $\text{P}(X=0)$.
Compute the probability that the price goes up on the first day and then down on the following four days.
Why is $\text{P}(X=1)$ different from the probability in the previous part? Compute and interpret $\text{P}(X=1)$.
Compute and interpret $\text{P}(X=2)$.
Suggest a general formula for the probability mass function of $X$.
Construct a table and plot to represent the distribution of $X$, and interpret it.

Consider a fixed number, $n$, of Bernoulli($p$) trials and let $X$ count the number of successes. The distribution of $X$ is defined to be the Binomial($n, p$) distribution.
A Binomial distribution is specified by two parameters:
- $n$ (an integer): the fixed number of Bernoulli trials, or the sample size
- $p$ (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 Binomial distribution with parameters $n$, a nonnegative integer, and $p\in[0, 1]$ if and only if its probability mass function is \[\begin{align*} p_{X}(x) & = \binom{n}{x} p^x (1-p)^{n-x}, & x=0, 1, 2, \ldots, n \end{align*}\]
- In the Binomial($n$, $p$) situation, there are $\binom{n}{x}$ S/F sequences that result in exactly $x$ successes—and thus $n-x$ failures—in $n$ trials,
- and each of these sequences has probability $p^x(1-p)^{n-x}$
The binomial coefficient (read “$n$ choose $x$”) \[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \] counts the number of success/failure sequences of length $n$ in which there are exactly $x$ successes. (Remember: by definition $0! = 1$.)

Example 10.2 Continuing Example 10.1.

Suggest a shortcut formula for $\text{E}(X)$.
Compute $\text{E}(X)$ using the distribution of $X$ from Example 10.1. Did the shortcut formula work?
Interpret $\text{E}(X)$.
Remember that we assumed $p=0.55$. If instead $p$ were 0.9, would $\text{Var}(X)$ be greater or less? What if $p=0.5$?

If $X$ has a Binomial($n$, $p$) distribution then \[\begin{align*} \text{E}(X) & = np\\ \text{Var}(X) & = np(1-p) \end{align*}\]

10.1 Exercises

Exercise 10.1 Continuing Example 10.1.

What does the random variable $5-X$ represent? What is its distribution?
Suppose that the price is currently $100 and each it either moves up $2 or down $2. Let $S$ be the stock price after 5 days. How does $S$ relate to $X$? Does $S$ have a Binomial distribution?
Recall that $X$ is the number of days on which the price goes up in the next five days. Suppose that $Y$ is the number of days on which the price goes up in the ten days after that (days 6-15). What is the distribution of $Y$? What does $X+Y$ represent and what is its distribution? (Continue to assume independence between days, with probability 0.55 of an up movement on any day.)

If $X$ and $Y$ are independent, $X$ has a Binomial($n_X$, $p$) distribution, and $Y$ has a Binomial($n_Y$, $p$) distribution then $X+Y$ has a Binomial($n_X+n_Y$, $p$)
A Binomial(1, $p$) distribution is also known as a Bernoulli($p$) distribution, taking a value of 1 with probability $p$ and 0 with probability $1-p$.
- For a single trial, a Bernoulli random variable indicates if there is a success (1) or failure (0) on the trial
If $X_1, X_2, \ldots, X_n$ are independent each with a Bernoulli($p$) distribution, then $X_1+\cdots+X_n$ has a Binomial($n, p$) distribution.
- The random variable $X_1+ X_2+ \ldots + X_n$ incrementally counts the total number of successes in the $n$ trials

Exercise 10.2 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.

Roll a die 20 times; $X$ is the number of times the die lands on an even number.
Roll a die 20 times; $X$ is the number of times the die lands on 6.
Roll a die until it lands on 6; $X$ is the total number of rolls.
Roll a die 20 times; $X$ is the sum of the numbers rolled.
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.)
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).
Randomly select a sample of 35 Cal Poly students; $X$ is the number of students in the sample who are CA residents.

Consider selecting a random sample of size $n$ from a population in which every individual is classified as “success” or “failure”. Suppose that the proportion of individuals in the population classified as success is $p$.
The individuals selected for the sample can be considered 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 the population proportion of success $p$
- The trials are independent
  - If sampling with replacement, or
  - If sampling without replacement but the population size is much larger than the sample size $n$ (which is usually the case in practice)
If $X$ counts the number of successes in a random sample of size $n$ from a large population with population proportion of success $p$ then $X$ has a Binomial($n, p$) distribution.

Exercise 10.3 Donny Dont is thoroughly confused about the distinction between a random variable and its distribution. Help him understand by by providing a simple concrete example of two different random variables $X$ and $Y$ that have the same distribution. Can you think of $X$ and $Y$ that have the same distribution but for which $\text{P}(X = Y) = 0$?