AP Statistics : How to find the mean of the sum of independent random variables

Study concepts, example questions & explanations for AP Statistics

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Example Questions

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

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

Possible Answers:

Correct answer:

Explanation:

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

Example Question #131 :Statistical Patterns And Random Phenomena

如果你有十个独立随机杂物bles, normally distributed with meanand variance, what is the distribution of the average of the random variables,

Possible Answers:

Normal distribution with with meanand variance.

Chi-square distribution withdegrees of freedom.

Normal distribution with meanand variance.

Normal distribution with meanand variance.

Correct answer:

Normal distribution with with meanand 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

Eachis, 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 meanand variance.

Example Question #1 :Measures Of Independent Random Variables

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

has meanand variance,

has meanand variance,

has meanand variance.

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

Possible Answers:

Correct answer:

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. Sohas 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 meanand variance.

This sum is a standard normal distribution.

The chance thatis 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:

Possible Answers:

Correct answer:

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 isand there are 10 melons.

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