Dominique answers a multiple choice questionnaire with 10 questions.
For each question, 4 answers are proposed, of which only one is correct.
Dominique answers randomly to each of the 10 questions by checking, for each question, exactly one box among the 4.
For each question, the probability that he answers correctly is therefore $\frac { 1 } { 4 }$.
We denote $X$ the random variable that counts the number of correct answers to this questionnaire.

\begin{enumerate}
  \item Determine the distribution followed by the random variable $X$ and give the parameters of this distribution.
  \item What is the probability that Dominique obtains exactly 5 correct answers? Round the result to $10 ^ { - 4 }$ near.
  \item Give the expectation of $X$ and interpret this result in the context of the exercise.
  \item We suppose in this question that a correct answer gives one point and an incorrect answer loses 0.5 points. The final score can therefore be negative.
\end{enumerate}

We denote $Y$ the random variable that gives the number of points obtained.
a. Calculate $P ( Y = 10 )$, give the exact value of the result.\\
b. From how many correct answers is Dominique's final score positive? Justify.\\
c. Calculate $P ( Y \leqslant 0 )$, give an approximate value to the nearest hundredth.\\
d. Show that $Y = 1.5 X - 5$.\\
e. Calculate the expectation of the random variable $Y$.