\textbf{Part A}

We consider a random variable $X$ that follows the exponential distribution with parameter $\lambda$ where $\lambda > 0$. We recall that, for every strictly positive real number $a$,
$$P(X \leqslant a) = \int_0^a \lambda \mathrm{e}^{-\lambda t} \,\mathrm{d}t$$
We propose to calculate the mathematical expectation of $X$, denoted $E(X)$, and defined by
$$E(X) = \lim_{x \to +\infty} \int_0^x \lambda t \mathrm{e}^{-\lambda t} \,\mathrm{d}t$$
We admit that the function $F$ defined on $\mathbb{R}$ by $F(t) = -\left(t + \frac{1}{\lambda}\right)\mathrm{e}^{-\lambda t}$ is an antiderivative on $\mathbb{R}$ of the function $f$ defined on $\mathbb{R}$ by $f(t) = \lambda t \mathrm{e}^{-\lambda t}$.

\begin{enumerate}
  \item Let $x$ be a strictly positive real number. Verify that
$$\int_0^x \lambda t \mathrm{e}^{-\lambda t} \,\mathrm{d}t = \frac{1}{\lambda}\left(-\lambda x \mathrm{e}^{-\lambda x} - \mathrm{e}^{-\lambda x} + 1\right)$$
  \item Deduce that $E(X) = \frac{1}{\lambda}$.
\end{enumerate}

\textbf{Part B}

The lifetime, expressed in years, of an electronic component can be modeled by a random variable denoted $X$ following the exponential distribution with parameter $\lambda$ where $\lambda > 0$.

\begin{enumerate}
  \item On the graph of appendix 2 (to be returned with the answer sheet):\\
a. Represent the probability $P(X \leqslant 1)$.\\
b. Indicate where the value of $\lambda$ can be read directly.
  \item We assume that $E(X) = 2$.\\
a. What does the value of the mathematical expectation of the random variable $X$ represent in the context of the exercise?\\
b. Calculate the value of $\lambda$.\\
c. Calculate $P(X \leqslant 2)$. Give the exact value then the value rounded to 0.01. Interpret this result.\\
d. Given that the component has already functioned for one year, what is the probability that its total lifetime is at least three years? Give the exact value.
\end{enumerate}

\textbf{Part C}

An electronic circuit is composed of two identical components numbered 1 and 2. We denote by $D_1$ the event ``component 1 fails before one year'' and we denote by $D_2$ the event ``component 2 fails before one year''.\\
We assume that the two events $D_1$ and $D_2$ are independent and that $P(D_1) = P(D_2) = 0,39$.\\
Two possible configurations are considered:

\begin{enumerate}
  \item When the two components are connected ``in parallel'', circuit A fails only if both components fail at the same time. Calculate the probability that circuit A fails before one year.
  \item When the two components are connected ``in series'', circuit B fails as soon as at least one of the two components fails. Calculate the probability that circuit B fails before one year.
\end{enumerate}