Each phone call by Ali consumes an amount of time that follows an exponential distribution with mean 5 minutes. The number of different phone calls Ali makes on any given day has a Poisson distribution with mean 3. Assume that a single call always falls within a single day (no calls continue past midnight).

Further, suppose that the number of phone calls that Ali makes on different days are independent, and that the lengths of the phone calls are also independent of each other. For simplicity, also assume that different phone calls never overlap and that there are 30 days in each given month.
Let 𝑋 be the total number of minutes Ali spends on the phone during one month.
Find 𝐄(𝑋) and 𝖵𝖺𝗋(𝑋).

Using the central limit theorem and a standard normal table or calculator, find the probability that the total number of phone calls Ali makes during an entire year (12 months of 30 days each) is between 1100 and 1200.
(Note that in this part of the question, you are asked about the number of phone calls, not the number of minutes.)

I got

E[X]=450
Var(X)=4500
(1100<=P<=1200)=0.2712

3. 0.2712

Var[∑i=1NXi] = E[N]Var[X1]+(E[X1])2Var(N).

=(5*3+25*5)*30=4500

Does that look right?

30*E_phones*E_minutes

Agree with mean = 450

As the variance is concerned: we CAN'T consider 30*Xi and so 30^2*var(Xi) because it does not make sense to multiply by 30 the 'randomness' of a single day. Actually, is it more appropriate to think at X as x1+x2+...+x30.

Given that all the days are independent, Var(x1+...+x30) = var(x1) + ... + var(x30) = 30 * var(xi) = 4500

Exp distribution mean = 5

mean = 1/lambda
variance = 1/lamdba^2
although the sum of exponential random variables over a day should be
mean=k/lambda variance = k/lambda^2 where k = 3 here

7200 is wrong

how do you get 0.2712?

1100 - 1080 / sqrt(1080) < P < 1200 - 1080 / sqrt(1080) ?

How did you guys find the variance?

I think the answer others have gotten for part 3 is slightly off: P(1100 ≤ P ≤ 1200) = P((1100 - 1080)/sqrt(1080) ≤ Z ≤ (1200-1080)/sqrt(1080)) = P(20/sqrt(1080) ≤ Z ≤ 120/sqrt(1080)) = P(Z ≤ 3.65148...) - P(Z ≤ 0.060858...) = 0.99987 - 0.72907 = 0.2708, not 0.2712. But if you're only giving 2 decimal places, it doesn't matter.