The hint is supposed to set me on track and I've rearranged to get $Y=\sum X_i$ but I don't see how to approximate the probability distribution. Similar questions on this site suggest the convolution of the PDF's of the variables but we haven't covered that in my course.
Q. The times that a cashier spends processing each person’s transaction are independent and identically distributed random variables with a mean of $\mu$ and a variance of $\sigma^2.$ Thus, if $X_i$ is the processing time for each transaction, $\operatorname{E}(X_i)= \mu$ and $\operatorname{Var}(X_i)= \sigma^2.$ Let $Y$ be the total processing time for 100 orders:
$Y = X_1 +X_2 + \dots +X_{100}.$
(a)What is the approximate probability distribution of $Y,$ the total processing time of 100 orders? Hint: $Y =100\bar X,$ where $\bar X = \frac 1 {100} \sum_{i=1}^{100} X_i$ is the sample mean.
Best Answer
Use the Central Limit Theorem. (I'm guessing you must have studied it recently.)
The sum (or mean) of your 100 iid random variables should be nearly normal, unless the distribution of each of them is very highly skewed. (You can't do a convolution because you're not given the distribution of each $X_i.$)
Also, $E(Y) = 100\mu$ and $Var(Y) = 100\sigma^2.$