\textbf{Exercise 5 — Candidates who have not followed the specialization course}\\
In this exercise, all requested probabilities will be rounded to $10 ^ { - 4 }$.\\
We study a model of automobile air conditioner composed of a mechanical module and an electronic module. If a module fails, it is replaced.

\textbf{Part A: Study of mechanical module failures}\\
An automobile maintenance company has found, through a statistical study, that the operating time (in months) of the mechanical module can be modeled by a random variable $D$ that follows a normal distribution with mean $\mu = 50$ and standard deviation $\sigma$:

\begin{enumerate}
  \item Determine the rounding to $10 ^ { - 4 }$ of $\sigma$ knowing that the statistical service indicates that $P ( D \geqslant 48 ) = 0,7977$.
\end{enumerate}

For the rest of this exercise, we will take $\sigma = 2,4$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Determine the probability that the operating time of the mechanical module is between 45 and 52 months.
  \item Determine the probability that the mechanical module of an air conditioner that has been operating for 48 months will continue to function for at least 6 more months.
\end{enumerate}

\textbf{Part B: Study of electronic module failures}\\
On the same air conditioner model, the automobile maintenance company has found that the operating time (in months) of the electronic module can be modeled by a random variable $T$ that follows an exponential distribution with parameter $\lambda$.

\begin{enumerate}
  \item Determine the exact value of $\lambda$, knowing that the statistical service indicates that $P ( 0 \leqslant T \leqslant 24 ) = 0,03$.
\end{enumerate}

For the rest of this exercise, we will take $\boldsymbol { \lambda } = 0,00127$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Determine the probability that the operating time of the electronic module is between 24 and 48 months.
  \item a. Prove that, for all positive real numbers $t$ and $h$, we have: $P _ { T \geqslant t } ( T \geqslant t + h ) = P ( T \geqslant h )$, that is, the random variable $T$ is memoryless.\\
b. The electronic module of the air conditioner has been operating for 36 months. Determine the probability that it will continue to function for the next 12 months.
\end{enumerate}

\textbf{Part C: Mechanical and electronic failures}\\
We admit that the events ( $D \geqslant 48$ ) and ( $T \geqslant 48$ ) are independent.\\
Determine the probability that the air conditioner does not fail before 48 months.

\textbf{Part D: Special case of a company garage}\\
A garage of the company has studied the maintenance records of 300 air conditioners over 4 years old. It finds that 246 of them have their mechanical module in working order for 4 years. Should this report call into question the result given by the company's statistical service, namely that $P ( D \geqslant 48 ) = 0,7977$? Justify the answer.