# AP Statistics : How to find standard deviation of a random variable

## Example Questions

### 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 #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
•  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

### 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?