Exercise 1 — Part B

We randomly choose a person who came to the multisports centre on a weekend. We denote $T_1$ the random variable giving their total waiting time in minutes before access to sports activities during Saturday and $T_2$ the random variable giving their total waiting time in minutes before access to sports activities during Sunday. We admit that:
\begin{itemize}
  \item $T_1$ follows a probability distribution with expectation $E(T_1) = 40$ and standard deviation $\sigma(T_1) = 10$;
  \item $T_2$ follows a probability distribution with expectation $E(T_2) = 60$ and standard deviation $\sigma(T_2) = 16$;
  \item the random variables $T_1$ and $T_2$ are independent.
\end{itemize}
We denote $T$ the random variable giving the total waiting time before access to sports activities over the two days, expressed in minutes. Thus we have $T = T_1 + T_2$.

\begin{enumerate}
  \item Determine the expectation $E(T)$ of the random variable $T$. Interpret the result in the context of the exercise.
  \item Show that the variance $V(T)$ of the random variable $T$ is equal to 356.
  \item Using the Bienaymé-Chebyshev inequality, show that, for a person randomly chosen among those who came to the multisports centre on a weekend, the probability that their total waiting time $T$ is strictly between 60 and 140 minutes is greater than 0.77.
\end{enumerate}