\section*{Exercise 1 (6 points)}

We study certain characteristics of a supermarket in a small town.

\section*{Part A - Preliminary Demonstration}
Let $X$ be a random variable that follows the exponential distribution with parameter 0.2.\\
Recall that the expectation of the random variable $X$, denoted $E(X)$, is equal to:
$$\lim_{x \rightarrow +\infty} \int_{0}^{x} 0.2t\, \mathrm{e}^{-0.2t} \mathrm{~d}t$$
The purpose of this part is to demonstrate that $E(X) = 5$.

\begin{enumerate}
  \item Let $g$ be the function defined on the interval $[0; +\infty[$ by $g(t) = 0.2t\,\mathrm{e}^{-0.2t}$.

We define the function $G$ on the interval $[0; +\infty[$ by $G(t) = (-t-5)\mathrm{e}^{-0.2t}$.\\
Verify that $G$ is a primitive of $g$ on the interval $[0; +\infty[$.
  \item Deduce that the exact value of $E(X)$ is 5.

Hint: you may use, without proving it, the following result:
$$\lim_{x \rightarrow +\infty} x\,\mathrm{e}^{-0.2x} = 0$$
\end{enumerate}

\section*{Part B - Study of the duration of a customer's presence in the supermarket}
A study commissioned by the supermarket manager makes it possible to model the duration, expressed in minutes, spent in the supermarket by a randomly chosen customer using a random variable $T$. This variable $T$ follows a normal distribution with expectation 40 minutes and standard deviation a positive real number denoted $\sigma$. Thanks to this study, it is estimated that $P(T < 10) = 0.067$.

\begin{enumerate}
  \item Determine an approximate value of the real number $\sigma$ to the nearest second.
  \item In this question, we take $\sigma = 20$ minutes. What is then the proportion of customers who spend more than one hour in the supermarket?
\end{enumerate}

\section*{Part C - Waiting time for payment}
This supermarket gives customers the choice to use automatic payment terminals alone or to go through a checkout managed by an operator.

\begin{enumerate}
  \item The waiting time at an automatic terminal, expressed in minutes, is modeled by a random variable that follows the exponential distribution with parameter $0.2\,\mathrm{min}^{-1}$.\\
a. Give the average waiting time for a customer at an automatic payment terminal.\\
b. Calculate the probability, rounded to $10^{-3}$, that the waiting time for a customer at an automatic payment terminal is greater than 10 minutes.
  \item The study commissioned by the manager leads to the following modeling:
\begin{itemize}
  \item among customers who chose to use an automatic terminal, 86\% wait less than 10 minutes;
  \item among customers using a checkout, 63\% wait less than 10 minutes.
\end{itemize}
We randomly choose a customer from the store and define the following events:\\
$B$: ``the customer pays at an automatic terminal'';\\
$\bar{B}$: ``the customer pays at a checkout with an operator'';\\
$S$: ``the customer's waiting time during payment is less than 10 minutes''.\\
A waiting time greater than ten minutes at a checkout with an operator or at an automatic terminal creates a negative perception of the store in the customer. The manager wants more than 75\% of customers to wait less than 10 minutes.\\
What is the minimum proportion of customers who must choose an automatic payment terminal for this objective to be achieved?
\end{enumerate}

\section*{Part D - Gift vouchers}
During payment, scratch cards, winning or losing, are distributed to customers. The number of cards distributed depends on the amount of purchases. Each customer receives one scratch card per 10~\euro{} of purchases.\\
For example, if the purchase amount is 58.64~\euro{}, then the customer receives 5 cards; if the amount is 124.31~\euro{}, the customer receives 12 cards.\\
Winning cards represent 0.5\% of the entire stock of cards. Furthermore, this stock is large enough to treat the distribution of a card as a draw with replacement.

\begin{enumerate}
  \item A customer makes purchases for an amount of 158.02~\euro{}.

What is the probability, rounded to $10^{-2}$, that they obtain at least one winning card?
  \item From what purchase amount, rounded to 10~\euro{}, is the probability of obtaining at least one winning card greater than 50\%?
\end{enumerate}