By how much does the mean and standard deviation change over a certain amount?

Question

The monthly profit of a small convenience store is a random variable with mean μ = 100 000 and standard deviation σ = 6 000. If we define Y to be the profit per year, assume that the monthly profits are independent and find:

10. The mean of Y
11. The standard deviation of Y

Do I simply multiply the mean and standard deviation of X by 12 for the profit per year?

Does that imply mean of Y is equal to 1, 200, 000 and standard deviation is 72, 000?

Answer 1

Hint:

Let $X$ and $Y$ be random variables. Then:

• $\mathbb E(X+Y)=\mathbb EX+\mathbb EY$ (and $\mathbb E(aX)=a\mathbb EX$). Linearity of expectation.
• If moreover $X$ and $Y$ are independent then $\text{Var}(X+Y)=\text{Var}X+\text{Var}Y$.

Apply this on $Y=Y_1+\cdots+Y_{12}$. If you know variance then you can deduce standard deviation and vice versa.