# AP Statistics : Measures of Independent Random Variables

## Example Questions

### Example Question #1 : How To Find The Mean Of The Sum Of Independent Random Variables

If is a random variable with a mean of and standard deviation of , what is the mean and standard deviation of ?

Explanation:

Remember how the mean and standard deviation of a random variable are affected when it is multiplied by a constant.

### Example Question #1 : How To Find The Mean Of The Sum Of Independent Random Variables

If you have ten independent random variables , normally distributed with mean  and variance , what is the distribution of the average of the random variables,

Normal distribution with mean  and variance .

Normal distribution with mean  and variance .

Normal distribution with with mean  and variance .

Chi-square distribution with  degrees of freedom.

Normal distribution with with mean  and variance .

Explanation:

Any linear combination of independent random variables is also normally distributed with the mean and variance depending on the weights on the random variables. The mean is additive in the sense that

Each  is , so the sum is equal to zero.

This means the sum of the average

is .

The variance satisfies

because of independence.

This means that the average is normally distributed with mean  and variance .

### Example Question #1 : How To Find The Mean Of The Sum Of Independent Random Variables

Suppose you have three independent normally distributed random variables, , such that

has mean  and variance ,

has mean  and variance ,

has mean  and variance .

What is the probability that the sum, , is less than ?

Explanation:

There is a relatively simple way of doing this problem. The sum of any set of independent normal random variables is also distributed normally. So  has a normal distribution. Now we can compute the mean and variance. The mean is additive:

Variance is also additive in some sense, when the random variables are independent:

Thus,  is normally distributed with mean  and variance .

This sum is a standard normal distribution.

The chance that  is thus , if we use a normal table.

### Example Question #1 : How To Find The Mean Of The Sum Of Independent Random Variables

An experiment is conducted on the watermelons that were grown on a small farm.  They want to compare the average weight of the melons grown this year to the average weight of last year's melons.  Find the mean of this year's watermelons using the following weights:

Explanation:

To find the mean you sum up all of your values then divide by the total amount of values.  The total sum of the weights is  and there are 10 melons.

### Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

A high school calculus exam is administered to a group of students. Upon grading the exam, it was found that the mean score was 95 with a standard deviation of 12. If one student's z score is 1.10, what is the score that she received on her test?

108.2

107.2

109.2

110.1

105.3

108.2

Explanation:

The z-score equation is given as: z = (X - μ) / σ, where X is the value of the element, μ is the mean of the population, and σ is the standard deviation. To solve for the student's test score (X):

X = ( z * σ) + 95 = ( 1.10 * 12) + 95 = 108.2.

### Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

and are independent random variables. If has a mean of  and standard deviation of  while variable has a mean of  and a standard deviation of , what are the mean and standard deviation of ?

Explanation:

First, find that has  and standard deviation .

Then find the mean and standard deviation of .

### Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

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

•  with
•  with
•  with
•  with

Compute the variance of the distribution.

Explanation:

The variance of a discrete random variable is computed as

for all the values of  that the random variable  can take.

First, we compute , which is the expected value. In this case, it is .

So we have

### Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

Clothes 4 Kids uses standard boxes to ship their clothing orders and the mean weight of the clothing packed in the boxes is  pounds. The standard deviation is  pounds. The mean weight of the boxes is  pound with a standard deviation of  pounds. The mean weight of the plastic packaging is  pounds per box, with a  pound standard deviation. What is the standard deviation of the weights of the packed boxes?