Let $P_k(t)$ denote the probability that $k$ small aircraft arrive during a $t$-hours period. Then
\[
P_k(t)=e^{-\alpha t}\cdot\frac{(\alpha t)^k}{k!}
\]
$\lambda=\alpha t$, $\alpha=8$.
Use $X_t$ to denote the number of small aircraft arrived during a $t$-hour period.
(a) For $k=5$, $t=1$, we have
\[
P(X_1=5)=P_5(1)=e^{-8}\cdot\frac{8^5}{5!}\approx 0.091604
\]
That is, the probability that exactly $5$ small aircraft arrive during a 1-hour period is 0.091604.
If we consider at least $5$ small aircraft will arrive during 1-hour period, then the probability of this event is
\[
\begin{split}
P(X_1\geqslant 5)&=\sum_{k=5}^{+\infty}P_k(1)=1-\sum_{k=0}^{4}P_k(1)\\
&=1-\sum_{k=0}^{4}e^{-8}\cdot\frac{8^k}{k!}\\
&\approx 1-0.099632\\
&=0.900368
\end{split}
\]
For the event that at least $10$ small aircraft,
\[
\begin{split}
P(X_1\geqslant 10)&=\sum_{k=10}^{+\infty}P_k(1)=1-\sum_{k=0}^{9}P_k(1)\\
&=1-\sum_{k=0}^{9}e^{-8}\cdot\frac{8^k}{k!}\\
&\approx 1-0.716624\\
&=0.283376
\end{split}
\]
(b)
For the $90$-min period. First $90$ minutes is equal to $1.5$ hour. That is, $t=1.5$. The rv is $X_{1.5}$
\[\lambda=\alpha t=8\times 1.5=12\]
Therefore, the expected value of $X_1$ is $\mu=E(X_{1.5})=\lambda=12$, and the variance of $X_{1.5}$ is $V(X_{1.5})=\lambda=12$. So, the standard deviation of $X_{1.5}$ is $\sigma=\sqrt{V(X_{1.5})}=\sqrt{12}\approx 3.464102$.
(c)
Let $t=2.5$, the rv is $X_{2.5}$. And $\lambda=\alpha t=8\times 2.5=20$.
Then the probability that at least $20$ small aircraft arrive during a $2\frac{1}{2}$-hour period is
\[
\begin{split}
P(X_{2.5}\geqslant 20)&=\sum_{k=20}^{+\infty}P_k(2.5)=1-\sum_{k=0}^{19}P_k(2.5)\\
&=1-\sum_{k=0}^{19}e^{-\alpha t}\frac{(\alpha t)^k}{k!}\\
&=1-\sum_{k=0}^{19}e^{-20}\cdot\frac{20^k}{k!}\\
&\approx 1-0.470257\\
&=0.529743
\end{split}
\]
The probability that at most $10$ arrive during this period is
\[
\begin{split}
P(X_{2.5}\leqslant 10)&=\sum_{k=0}^{10}P_k(2.5)\\
&=\sum_{k=0}^{10}e^{-\alpha t}\frac{(\alpha t)^k}{k!}\\
&=\sum_{k=0}^{10}e^{-20}\cdot\frac{20^k}{k!}\\
&\approx 0.010812
\end{split}
\]
