Answer:
a. X is the number of adults in America that need to be surveyed until finding the first one that will watch the Super Bowl.
b. X can take any integer that is greater than or equal to 1. .
c. .
d. .
e. .
f. .
Step-by-step explanation:
<h3>a.</h3>In this setting, finding an adult in America that will watch the Super Bowl is a success. The question assumes that the chance of success is constant for each trial. The question is interested in the number of trials before the first success. Let X be the number of adults in America that needs to be surveyed until finding the first one who will watch the Super Bowl.
<h3>b.</h3>It takes at least one trial to find the first success. However, there's rare opportunity that it might take infinitely many trials. Thus, X may take any integer value that is greater than or equal to one. In other words, X can be any positive integer: .
There are two discrete distributions that may model X:
- The geometric distribution. A geometric random variable measures the number of trials before the first success. This distribution takes only one parameter: the chance of success on each trial.
- The negative binomial distribution. A negative binomial random variable measures the number of trials before the r-th success. This distribution takes two parameters: the number of successes
and the chance of success on each trial
.
(note that
) is equivalent to
. However, in this question the distribution of
takes two parameters, which implies that
shall follow the negative binomial distribution rather than the geometric distribution. The probability of success on each trial is
.
.
The expected value of a negative binomial random variable is equal to the number of required successes over the chance of success on each trial. In other words,
.
.
Some calculators do not come with support for the negative binomial distribution. There's a walkaround for that as long as the calculator supports the binomial distribution. The r-th success occurs on the n-th trial translates to (r-1) successes on the first (n-1) trials, plus another success on the n-th trial. Find the chance of (r-1) successes in the first (n-1) trials and multiply that with the chance of success on the n-th trial.
<h3>f.</h3>.