# AP Statistics : Random Variables

## Example Questions

### Example Question #3 : How To Find Mean Of A Random Variable

There are  collectable coins in a bag.  are  ounces,  are  ounces,  are  ounces, and  are  ounces. If one coin is randomly selected, what is the mean possible weight in ounces?

Explanation:

We are required to find the mean outcome where the probability of each possible result varies--the random/weighted mean.

First, multiply each possible outcome by the probability of that outcome occurring.

### Example Question #1 : How To Find Mean Of A Random Variable

A basketball player makes  of his three-point shots. If he takes  three-point shots each game, how many points per game does he score from three-point range?

Explanation:

First convert .

The player's three-point shooting follows a binomial distribution with  and .

On average, he thus makes  three-point shots per game.

This means he averages 12 points per game from three-point range if he tries to make 10 three-pointers per game.

### Example Question #1 : How To Find Mean Of A Random Variable

Tim samples the average plant height of potato plants for his science class and finds the following distribution (in inches):

Which of the following is/are true about the data?

i: the mode is

ii: the mean is

iii: the median is

iv: the range is

i, ii & iii

ii, iii & iv

i & iii

all of the above

i & ii

i & ii

Explanation:

Analyzing the data, there are more 6s than anything else (mode), the median is between  and     , the mean is , and the range is

### Example Question #1 : How To Find Standard Deviation Of A Random Variable

Robert's work schedule for next week will be released today.  Robert will work either 45, 40, 25, or 12 hours.  The probabilities for each possibility are listed below:

45 hours: 0.3

40 hours: 0.2

25 hours: 0.4

12 hours: 0.1

What is the standard deviation of the possible outcomes?

Explanation:

There are four steps to finding the standard deviation of random variables. First, calculate the mean of the random variables. Second, for each value in the group (45, 40, 25, and 12), subtract the mean from each and multiply the result by the probability of that outcome occurring. Third, add the four results together. Fourth, find the square root of the result.

### Example Question #1 : How To Find Standard Deviation Of A Random Variable

We have two independent, normally distributed random variables  and  such that  has mean  and variance  and  has mean  and variance . What is the probability distribution of the difference of the random variables, ?

Normal distribution with mean  and variance .

Normal distribution with mean  and variance .

Normal distribution with mean  and variance .

Normal distribution with mean  and variance .

Normal distribution with mean  and variance .

Explanation:

The mean for any set of random variables is additive in the sense that

The difference is also additive, so we have

This means the mean of  is

The variance is additive when the random variables are independent, which they are in this case. But it's additive in the sense that for any real numbers  (even when negative), we have

.

So for this difference, we have

.

So the mean and variance are  and , respectively. In addition to that,  is normally distributed because the sum or difference of any set of independent normal random variables is also normally distributed.

### Example Question #1 : How To Find Standard Deviation Of A Random Variable

If  and  are two independent random variables with  and , what is the standard deviation of the sum,

Explanation:

If the random variables are independent, the variances are additive in the sense that

.

So then the variance of the sum is

.

The standard deviation is the square root of the variance, so we have

.

### Example Question #1 : How To Find Standard Deviation Of A Random Variable

Consider the discrete random variable  that takes the following values with the corresponding probabilities:

•  with
•  with
•  with

Compute the probability

Explanation:

This probability is simple to compute:

We want to add the probability that X is greater or equal to two. This means the probability that X=2 or X=3.

Adding the necessary probabilities we arrive at the solution.

### Example Question #2 : How To Find Standard Deviation Of A Random Variable

Consider the discrete random variable  that takes the following values with the corresponding probabilities:

•  with
•  with
•  with
•  with

Compute the expected value of the distribution.

Explanation:

The expected value is computed as

for any values of x that the random variable takes.

So we have

### Example Question #1 : How To Find Standard Deviation Of A Random Variable

The average number of calories in a Lick Yo' Lips lollipop is , with a standard deviation of . The calories per lollipop are normally distributed, so what percent of lollipops have more than  calories?

Explanation:

The random variable  number of calories per lollipop, so the answer is

or