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

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Question

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

Answer

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

$E(cX)=c\cdot E(X)$

$\mu = 2\cdot 10 = 20$

$\sigma=\sqrt{c^2\cdot 5}$

$\sigma = \sqrt{2^2 \cdot 5} = \sqrt{20}$

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Question

If you have ten independent random variables $\small X_1,X_2,...,X_{10}$ , normally distributed with mean $\small 0$ and variance $\small 1$ , what is the distribution of the average of the random variables, $\small \small \frac{X_1+X_2+...+X_{10}}{10}?$

Answer

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

$\small mean\left(\frac{X_1+...+X_{10}}{10}\right)=\frac{1}{10}\left(mean(X_1)+...+mean(X_{10})\right)$

Each $\small mean(X_i)$ is $\small 0$ , so the sum is equal to zero.

This means the sum of the average

$\small \small \small \frac{X_1+X_2+...+X_{10}}{10}$ is $\small 0$ .

The variance satisfies

$Var\left(\frac{X_1+X_2+...+X_{10}}{10}\right)=\frac{1}{10^2}Var(X_1+...+X_{10})$

$\small =\frac{1}{100}(Var(X_1)+...Var(X_{10}))=\frac{10}{100}=\frac{1}{10}$ because of independence.

This means that the average is normally distributed with mean $\small 0$ and variance $\small \frac{1}{10}$ .

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Question

Suppose you have three independent normally distributed random variables, $\small X_1,X_2,X_3$ , such that

$\small X_1$ has mean $\small \frac{1}{4}$ and variance $\small \frac{1}{2}$ ,

$\small X_2$ has mean $\small -1$ and variance $\small \frac{3}{8}$ ,

$\small X_3$ has mean $\small \frac{3}{4}$ and variance $\small \frac{1}{8}$ .

What is the probability that the sum, $\small X_1+X_2+X_3$ , is less than $\small -1$ ?

Answer

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 $\small X_1+X_2+X_3$ has a normal distribution. Now we can compute the mean and variance. The mean is additive:

$\small mean(X_1+X_2+X_3)=mean(X_1)+mean(X_2)+mean(X_3)$

$\small \small =\frac{1}{4}-1+\frac{3}{4}=0$

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

$\small \small Var(X_1+X_2+X_3)=Var(X_1)+Var(X_2)+Var(X_3)$

$\small =\frac{1}{2}+\frac{3}{8}+\frac{1}{8}=1$

Thus, $\small X_1+X_2+X_3$ is normally distributed with mean $\small 0$ and variance $\small 1$ .

This sum is a standard normal distribution.

The chance that $\small X_1+X_2+X_3<-1$ is thus $\small 0.1587$ , if we use a normal table.

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Question

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:

Answer

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. $60.5\div10=60.5lbs$

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