Sum of a exponential distribution

Question

I am working on a math question that I don't understand how to solve.

The question is as follows: Consider a reception where the time Xi is an Exp(λ) distributed variable where λ = 10/h. Assume that Xi is independent and that there is no delay between customers. Determine the approximate distribution for the time it takes to serve 100 customers.

A: N(10,1/10) B: N(10,1) C: N(10,10) D: N(100,10)

I thought the sum of an exponential variable was a variable with gamma distributed random variable. Not sure how to solve this question to be honest. Help appreciated :).

Answer 1

Hint:

• You are expected to use the Central Limit Theorem to justify a normal approximation
• What are the mean and variance for the service time of $1$ person?
• What are the mean and variance for the sum of the service times of $100$ people?