I have been trying to solve the following simple example, but there is something I do not understand:
Consider a customer reception where the time $Xi$ it takes to serve a customer is $Exp (λ)$ -distributed with $λ = 10h^{-1}$. Assume that all $Xi$ are independent and customer $i+1$ is served immediately after the customer $i$ is served. Determine the approximate distribution for the total time it takes to serve the 100 first customers.
The answer to this question is $N(10,1)$. However, I do not understand how did they reach that. I was thinking of using the central limit theorem in this case, but I am uncertain what does $h$ in $λ$ above represent?
Any help would be appreciated.
$10h^{-1}$ represents a rate of ten customers per hour. So the time unit is hours
For one customer and an exponential distribution, the mean is $\lambda^{-1} = 0.1$ and the variance is $\lambda^{-2} = 0.01$
So $100$ independent customers, the mean is $100\times 0.1=10$ and the variance is $100 \times 0.01 = 1$
You can apply the Central Limit Theorem to give a normal approximation; the exact distribution is a Gamma distribution with shape parameter $100$ and rate parameter $10$. The two distributions have the same mean and variance and similar densities as shown below, though the Gamma distribution (blue) is very slightly right skewed and has a mode of $9.9$ compared with the symmetric normal (red) with mode of $10$