Gamma and Exponential distribution question?

Question

The working time of one bank has an exponential distribution with a parameter λ=0.1 (in minutes). You came in the bank, but there were already 35 people before you. What's the probability that all of them will be done in 45 minutes?

Answer 1

This is related to Poisson process (look it up). The probability that they will be done over time $T$ is the probability that $N \ge 35$ where $N \sim Poisson(\lambda T)$. You can calculate that using software.

Also, you can apply Normal approximation, $Z \approx (N - \lambda T)/\sqrt{\lambda T}$.

Note: here, $\lambda$ is the rate of service (people per minute).

Answer 2

I would say that

$P(X\le 45)=P(\displaystyle\sum_1^kx_i\le45)=Erlang(k,\lambda)=\Gamma(\lambda, k);\,\lambda=0.1,\, k = 35$