The curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ given in appendix 1 are the graphical representations, in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ), of two functions $f$ and $g$ defined on $[ 0 ; + \infty [$.\\
We consider the points $\mathrm { A } ( 0,5 ; 1 )$ and $\mathrm { B } ( 0 ; - 1 )$ in the coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ).\\
We know that O belongs to $\mathscr { C } _ { f }$ and that the line (OA) is tangent to $\mathscr { C } _ { f }$ at point O.

\begin{enumerate}
  \item We assume that the function $f$ is written in the form $f ( x ) = ( a x + b ) \mathrm { e } ^ { - x ^ { 2 } }$ where $a$ and $b$ are real numbers. Determine the exact values of the real numbers $a$ and $b$, detailing the approach.\\
From now on, we consider that $\boldsymbol { f } ( \boldsymbol { x } ) = \mathbf { 2 } \boldsymbol { x } \mathrm { e } ^ { - \boldsymbol { x } ^ { \mathbf { 2 } } }$ for all $\boldsymbol { x }$ belonging to $[ \mathbf { 0 } ; + \infty [$
  \item a. We will admit that, for all real $x$ strictly positive, $f ( x ) = \frac { 2 } { x } \times \frac { x ^ { 2 } } { \mathrm { e } ^ { x ^ { 2 } } }$.

Calculate $\lim _ { x \rightarrow + \infty } f ( x )$.\\
b. Draw up, justifying it, the table of variations of the function $f$ on $[ 0 ; + \infty [$.
  \item The function $g$ whose representative curve $\mathscr { C } _ { g }$ passes through the point $\mathrm { B } ( 0 ; - 1 )$ is a primitive of the function $f$ on $[ 0 ; + \infty [$.\\
a. Determine the expression of $g ( x )$.\\
b. Let $m$ be a strictly positive real number.

Calculate $I _ { m } = \int _ { 0 } ^ { m } f ( t ) \mathrm { d } t$ as a function of $m$.\\
c. Determine $\lim _ { m \rightarrow + \infty } I _ { m }$.
  \item a. Justify that $f$ is a probability density function on $[ 0 ; + \infty [$.\\
b. Let $X$ be a continuous random variable that admits the function $f$ as its probability density function. Justify that, for all real $x$ in $[ 0 ; + \infty [$,\\
$P ( X \leqslant x ) = g ( x ) + 1$.\\
c. Deduce the exact value of the real number $\alpha$ such that $P ( X \leqslant \alpha ) = 0,5$.\\
d. Without using an approximate value of $\alpha$, construct in the coordinate system of appendix 1 the point with coordinates ( $\alpha ; 0$ ) leaving the construction lines visible.\\
Then shade the region of the plane corresponding to $P ( X \leqslant \alpha )$.
\end{enumerate}