Answer

Interarrival times is exponentially distributed with mean \(E[X] = 10 = 1/\lambda\). Hence \(\lambda = 0.1\).

1. For part 1, we need at least 10 requests within one hour. Implies that the sum of the first 10 interarrival times should be less than 60 minutes.

\[ \begin{aligned} P(T_{1} + \cdots + T_{10} < 60) \end{aligned} \]

We know that the sum of independent exponential variables is Gamma distribution. Hence,

\[ \begin{aligned} P(Gamma(10, 0.1) < 60) = 0.08392 \end{aligned} \]

2. Similar to the above part, we need the sum of first 10 interarrival times to be more than two hours

\[ \begin{aligned} P(T_{1} + \cdots + T_{10} \geq 120) = P(Gamma(10, 0.1) \geq 120) = 0.24239 \end{aligned} \]