To promote the organic products of his store, a store manager decides to organize a game that consists, for a customer, of filling a basket with a certain mass of apricots from organic farming. It is announced that the customer wins the contents of the basket if the mass of apricots deposited is between 3.2 and 3.5 kilograms.

The mass of fruit in kg, placed in the basket by customers, can be modeled by a random variable $X$ following the probability distribution with density $f$ defined on the interval $[3; 4]$ by:
$$f(x) = \frac{2}{(x-2)^2}$$

Reminder: a probability density function on the interval $[a; b]$ is any function $f$ defined, continuous and positive on $[a; b]$, such that the integral of $f$ over $[a; b]$ is equal to 1.

\begin{enumerate}
  \item Verify that the function $f$ previously defined is indeed a probability density function on the interval $[3;4]$.
  \item The store announces: ``One customer in three wins the basket!''. Is this announcement accurate?
  \item The purpose of this question is to calculate the mathematical expectation $\mathrm{E}(X)$.
\end{enumerate}