5 points

The probabilities requested should be rounded to 0.01.\\
A shopkeeper has just equipped himself with an Italian ice cream dispenser.

\begin{enumerate}
  \item The duration, in months, of operation without breakdown of his Italian ice cream dispenser is modelled by a random variable $X$ which follows an exponential distribution with parameter $\lambda$ where $\lambda$ is a strictly positive real number (recall that the density function $f$ of the exponential distribution is given on $\left[ 0 ; + \infty \left[ \text{ by } f ( x ) = \lambda \mathrm { e } ^ { - \lambda x } \right. \right.$).
\end{enumerate}

The seller of the device assures that the average duration of operation without breakdown of this type of dispenser, that is to say the mathematical expectation of $X$, is 10 months.\\
a. Justify that $\lambda = 0.1$.\\
b. Calculate the probability that the Italian ice cream dispenser has experienced no breakdown during the first six months.\\
c. Given that the dispenser has experienced no breakdown during the first six months, what is the probability that it experiences none until the end of the first year? Justify.\\
d. The shopkeeper will replace his Italian ice cream dispenser after a time $t$, expressed in months, which verifies that the probability of the event ( $X > t$ ) is equal to 0.05.\\
Determine the value of $t$ rounded to the nearest integer.\\
2. The dispenser manual specifies that the dispenser provides Italian ice creams whose mass is between 55 g and 65 g.\\
Consider the random variable $M$ representing the mass, in grams, of a distributed ice cream. It is admitted that $M$ follows the normal distribution with expectation 60 and standard deviation 2.5.\\
a. Calculate the probability that the mass of an Italian ice cream chosen at random from those distributed is between 55 g and 65 g.\\
b. Determine the largest value of $m$, rounded to the nearest gram, such that the probability $P ( M \geqslant m )$ is greater than or equal to 0.99.\\
3. The Italian ice cream dispenser allows you to choose only one of two flavours: vanilla or strawberry. To better manage his purchases of raw materials, the shopkeeper makes the hypothesis that there will be in proportion two vanilla ice cream buyers for one strawberry ice cream buyer. On the first day of use of his dispenser, he observes that out of 120 consumers, 65 chose vanilla ice cream.\\
For what mathematical reason could he doubt his hypothesis? Justify.