Let $X$ be a random variable that follows the exponential distribution with parameter $\lambda$, where $\lambda$ is a given strictly positive real number.\\
We recall that the probability density of this distribution is the function $f$ defined on $[ 0 ; + \infty [$ by
$$f ( x ) = \lambda \mathrm { e } ^ { - \lambda x } .$$

a. Let $c$ and $d$ be two real numbers such that $0 \leqslant c < d$.

Prove that the probability $P ( c \leqslant X \leqslant d )$ satisfies
$$P ( c \leqslant X \leqslant d ) = \mathrm { e } ^ { - \lambda c } - \mathrm { e } ^ { - \lambda d } .$$

b. Determine a value of $\lambda$ to $10 ^ { - 3 }$ near such that the probability $P ( X > 20 )$ is equal to 0.05.\\
c. Give the expectation of the random variable $X$

\textbf{In the rest of the exercise we take $\boldsymbol { \lambda } = \mathbf { 0 , 1 5 }$.}

d. Calculate $P ( 10 \leqslant X \leqslant 20 )$.\\
e. Calculate the probability of the event $( X > 18 )$.\\
2. Let $Y$ be a random variable that follows the normal distribution with expectation 16 and standard deviation 1.95.\\
a. Calculate the probability of the event $( 20 \leqslant Y \leqslant 21 )$.\\
b. Calculate the probability of the event $( Y < 11 ) \cup ( Y > 21 )$.