Exercise 5 — Candidates who have not followed the specialization course In this exercise, all requested probabilities will be rounded to $10 ^ { - 4 }$. We study a model of automobile air conditioner composed of a mechanical module and an electronic module. If a module fails, it is replaced.
Part A: Study of mechanical module failures An automobile maintenance company has found, through a statistical study, that the operating time (in months) of the mechanical module can be modeled by a random variable $D$ that follows a normal distribution with mean $\mu = 50$ and standard deviation $\sigma$:

1. Determine the rounding to $10 ^ { - 4 }$ of $\sigma$ knowing that the statistical service indicates that $P ( D \geqslant 48 ) = 0,7977$.

For the rest of this exercise, we will take $\sigma = 2,4$.

1. Determine the probability that the operating time of the mechanical module is between 45 and 52 months.
2. Determine the probability that the mechanical module of an air conditioner that has been operating for 48 months will continue to function for at least 6 more months.

Part B: Study of electronic module failures On the same air conditioner model, the automobile maintenance company has found that the operating time (in months) of the electronic module can be modeled by a random variable $T$ that follows an exponential distribution with parameter $\lambda$.

1. Determine the exact value of $\lambda$, knowing that the statistical service indicates that $P ( 0 \leqslant T \leqslant 24 ) = 0,03$.

For the rest of this exercise, we will take $\boldsymbol { \lambda } = 0,00127$.

1. Determine the probability that the operating time of the electronic module is between 24 and 48 months.
2. a. Prove that, for all positive real numbers $t$ and $h$, we have: $P _ { T \geqslant t } ( T \geqslant t + h ) = P ( T \geqslant h )$, that is, the random variable $T$ is memoryless. b. The electronic module of the air conditioner has been operating for 36 months. Determine the probability that it will continue to function for the next 12 months.

Part C: Mechanical and electronic failures We admit that the events ( $D \geqslant 48$ ) and ( $T \geqslant 48$ ) are independent. Determine the probability that the air conditioner does not fail before 48 months.
Part D: Special case of a company garage
A garage of the company has studied the maintenance records of 300 air conditioners over 4 years old. It finds that 246 of them have their mechanical module in working order for 4 years. Should this report call into question the result given by the company's statistical service, namely that $P ( D \geqslant 48 ) = 0,7977$? Justify the answer.

In this exercise, we study some characteristic quantities of the operation of parking lots in a city. Throughout the exercise, probabilities will be given with a precision of $10 ^ { - 4 }$.

Parts A, B, and C are independent

Part A - Waiting time to enter an underground parking lot

The waiting time is defined as the time that elapses between the moment the car arrives at the parking entrance and the moment it passes through the parking entrance barrier. The following table presents observations made over one day.

Waiting time in minutes	$[ 0 ; 2 [$	$[ 2 ; 4 [$	$[ 4 ; 6 [$	$[ 6 ; 8 [$
Number of cars	75	19	10	5

1. Propose an estimate of the average waiting time for a car at the parking entrance.
2. We decide to model this waiting time by a random variable $T$ following an exponential distribution with parameter $\lambda$ (expressed in minutes). a. Justify that we can choose $\lambda = 0.5 \mathrm {~min}$. b. A car arrives at the parking entrance. What is the probability that it takes less than two minutes to pass through the barrier? c. A car has been waiting at the parking entrance for one minute. What is the probability that it passes through the barrier in the next minute?

Part B - Duration and parking rates in this underground parking lot

Once parked, the parking duration of a car is modeled by a random variable $D$ that follows a normal distribution with mean $\mu = 70 \mathrm {~min}$ and standard deviation $\sigma = 30 \mathrm {~min}$.

1. a. What is the average parking duration for a car? b. A motorist enters and parks in the parking lot. What is the probability that their parking duration exceeds two hours? c. To the nearest minute, what is the maximum parking time for at least $99 \%$ of cars?
2. The parking duration is limited to three hours. The table gives the rate for the first hour and each additional hour is charged at a single rate. Any hour started is charged in full.

\begin{tabular}{ c } Parking

duration

& Less than 15 min & Between 15 min and 1 h &

Additional

hour

\hline Rate in euros & Free & 3.5 & $t$ \hline \end{tabular}
Determine the rate $t$ for the additional hour that the parking manager must set so that the average parking price for a car is 5 euros.

Part C - Waiting time to park in a city center parking lot

The parking duration of a car in a city center parking lot is modeled by a random variable $T ^ { \prime }$ that follows a normal distribution with mean $\mu ^ { \prime }$ and standard deviation $\sigma ^ { \prime }$. It is known that the average parking time in this lot is 30 minutes and that $75 \%$ of cars have a parking time less than 37 minutes. The parking manager aims for the objective that $95 \%$ of cars have a parking time between 10 and 50 minutes. Is this objective achieved?

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.

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

A continuous random variable $X$ has a range of $0 \leqq X \leqq 3$, and the graph of its probability density function is as follows. When $\mathrm { P } ( m \leqq X \leqq 2 ) = \mathrm { P } ( 2 \leqq X \leqq 3 )$, what is the value of $m$? (Here, $0 < m < 2$.) [3 points]
(1) $\frac { \sqrt { 2 } } { 2 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) 1
(4) $\sqrt { 2 }$
(5) $\sqrt { 3 }$

For two positive numbers $a , b$, a continuous random variable $X$ has a range of $0 \leqq X \leqq a$, and the graph of the probability density function is as shown. When $\mathrm { P } \left( 0 \leqq X \leqq \frac { a } { 2 } \right) = \frac { b } { 2 }$, find the value of $a ^ { 2 } + 4 b ^ { 2 }$. [4 points]

A continuous random variable $X$ has a range of $0 \leqq X \leqq 3$, and the probabilities $\mathrm { P } ( X \leqq 1 )$ and $\mathrm { P } ( X \leqq 2 )$ are the two roots of the quadratic equation $6 x ^ { 2 } - 5 x + 1 = 0$. What is the value of the probability $\mathrm { P } ( 1 < X \leqq 2 )$? [3 points]
(1) $\frac { 1 } { 12 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 5 } { 12 }$

A continuous random variable $X$ has a range of $0 \leqq X \leqq 4$, and the graph of the probability density function of $X$ is as shown in the figure. Find the value of $100 \mathrm { P } ( 0 \leqq X \leqq 2 )$. [4 points]

A continuous random variable $X$ defined on the interval $[ 0,1 ]$ has probability density function $f ( x )$. The mean of $X$ is $\frac { 1 } { 4 }$ and $\int _ { 0 } ^ { 1 } ( a x + 5 ) f ( x ) d x = 10$. Find the value of the constant $a$. [4 points]

A continuous probability density function is defined on the closed interval $[ 0 , a ]$ for a random variable $X$. When the random variable $X$ satisfies the following conditions, what is the value of the constant $k$? [4 points] (가) For all $x$ where $0 \leq x \leq a$, $\mathrm { P } ( 0 \leq X \leq x ) = k x ^ { 2 }$. (나) $\mathrm { E } ( X ) = 1$
(1) $\frac { 9 } { 16 }$
(2) $\frac { 4 } { 9 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 9 }$
(5) $\frac { 1 } { 16 }$

For a continuous random variable $X$ that takes all real values in the interval $[ 0,3 ]$, the graph of the probability density function of $X$ is shown in the figure. If $\mathrm { P } ( 0 \leq X \leq 2 ) = \frac { q } { p }$, find the value of $p + q$. (Here, $k$ is a constant, and $p$ and $q$ are coprime natural numbers.) [4 points]

For a continuous random variable $X$ that takes all real values in the closed interval $[ 0,1 ]$, the probability density function is $$f ( x ) = k x \left( 1 - x ^ { 3 } \right) \quad ( 0 \leq x \leq 1 )$$ Find the value of $24 k$. (Here, $k$ is a constant.) [3 points]

A continuous random variable $X$ has range $0 \leq X \leq 2$, and the graph of the probability density function of $X$ is shown in the figure. What is the value of $\mathrm { P } \left( \frac { 1 } { 3 } \leq X \leq a \right)$? (Here, $a$ is a constant.) [3 points] [Figure]
(1) $\frac { 11 } { 16 }$
(2) $\frac { 5 } { 8 }$
(3) $\frac { 9 } { 16 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 7 } { 16 }$

Two continuous random variables $X$ and $Y$ have ranges $0 \leq X \leq 6$ and $0 \leq Y \leq 6$, with probability density functions $f ( x )$ and $g ( x )$ respectively. The graph of the probability density function $f ( x )$ of random variable $X$ is shown in the figure.
For all $x$ with $0 \leq x \leq 6$, $$f ( x ) + g ( x ) = k \text{ (where } k \text{ is a constant)}$$
When $\mathrm { P } ( 6 k \leq Y \leq 15 k ) = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

A continuous random variable $X$ has a range of $0 \leq X \leq a$, and the graph of the probability density function of $X$ is as shown in the figure. When $\mathrm { P } ( X \leq b ) - \mathrm { P } ( X \geq b ) = \frac { 1 } { 4 }$ and $\mathrm { P } ( X \leq \sqrt { 5 } ) = \frac { 1 } { 2 }$, what is the value of $a + b + c$? (Here, $a$, $b$, and $c$ are constants.) [4 points]
(1) $\frac { 11 } { 2 }$
(2) 6
(3) $\frac { 13 } { 2 }$
(4) 7
(5) $\frac { 15 } { 2 }$

The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Is the application $u \longmapsto \dfrac{h(u)}{u+x}$ integrable on $\mathbb{R}_{+}$?

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Show that
$$\mathbb{E}(X^{2}) = 2 \int_{0}^{+\infty} t \mathbb{P}(|X| \geqslant t) \, dt$$
You may denote $X^{2}(\Omega) = \{y_{1}, \ldots, y_{n}\}$ with $0 \leqslant y_{1} < y_{2} < \cdots < y_{n}$.

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Show that the second moment of $X$ is less than or equal to $\frac{a}{b}$.

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$ and $F_n(x) = \mathbb{P}(Y_n \leqslant x)$.
Show $$\forall n \in \mathbb{N}^{\star}, \forall x \in D_n, \quad F_n(x) = x + \frac{1}{2^n}.$$

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$ and $G_n(x) = \mathbb{P}(Y_n < x)$.
Show $$\forall n \in \mathbb{N}^{\star}, \forall x \in D_n, \quad G_n(x) = x.$$

Let $\left(f_n\right)_{n \geqslant 1}$ be a sequence of functions from $\mathbb{N}^*$ to $\mathbb{R}$ such that, for all $x \in \mathbb{N}^*$, the sequence $\left(f_n(x)\right)_{n \geqslant 1}$ converges to a real number $f(x)$ as $n$ tends to $+\infty$. We assume that there exists a function $h : \mathbb{N}^* \rightarrow [0, +\infty[$ such that $h(X)$ has finite expectation and such that $\left|f_n(m)\right| \leqslant h(m)$ for all $m$ and $n$ in $\mathbb{N}^*$. Justify that $E(f(X))$ has finite expectation and show that $$\lim_{n \rightarrow +\infty} E\left(f_n(X)\right) = E(f(X)).$$

We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Show successively that $Y ^ { 2 }$ and $| Y | ^ { 3 }$ have finite expectation, and that
$$\mathrm { E } \left( Y ^ { 2 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 1 / 2 } \quad \text { then } \quad \mathrm { E } \left( | Y | ^ { 3 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$

We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Show, for all real $u$, the inequality
$$\left| e ^ { i u } - 1 - i u + \frac { u ^ { 2 } } { 2 } \right| \leq \frac { | u | ^ { 3 } } { 6 }$$
Deduce that for all real $\theta$,
$$\left| \Phi _ { Y } ( \theta ) - 1 + \frac { \mathbf { E } \left( Y ^ { 2 } \right) \theta ^ { 2 } } { 2 } \right| \leq \frac { | \theta | ^ { 3 } } { 3 } \left( \mathbf { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$

We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Conclude that for all real $\theta$,
$$\left| \Phi _ { Y } ( \theta ) - \exp \left( - \frac { \mathbf { E } \left( Y ^ { 2 } \right) \theta ^ { 2 } } { 2 } \right) \right| \leq \frac { | \theta | ^ { 3 } } { 3 } \left( \mathbf { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 } + \frac { \theta ^ { 4 } } { 8 } \mathbf { E } \left( Y ^ { 4 } \right)$$

Given a real $t > 0$, we set
$$m _ { t } : = S _ { 1,1 } ( t ) \quad \text { and } \quad \sigma _ { t } : = \sqrt { S _ { 2,1 } ( t ) }$$
Given reals $t > 0$ and $u$, we set
$$\zeta ( t , u ) = \exp \left( i \frac { u } { \sigma _ { t } } \left( m _ { t } - \frac { \pi ^ { 2 } } { 6 t ^ { 2 } } \right) \right) \quad \text { and } \quad j ( t , u ) = \zeta ( t , u ) h \left( t , \frac { u } { \sigma _ { t } } \right)$$
Show that $\sigma _ { t } \sim \frac { \pi } { \sqrt { 3 } t ^ { 3 / 2 } }$ as $t$ tends to $0 ^ { + }$. Deduce from this that, for all real $u$,
$$j ( t , u ) \underset { t \rightarrow 0 ^ { + } } { \longrightarrow } e ^ { - u ^ { 2 } / 2 }$$

Show that there exists a real $\alpha > 0$ such that
$$\forall \theta \in [ - \pi , \pi ] , 1 - \cos \theta \geq \alpha \theta ^ { 2 }$$
Using question $9 \triangleright$, deduce from this that there exist three real numbers $t _ { 0 } > 0 , \beta > 0$ and $\gamma > 0$ such that, for all $\left. t \in ] 0 , t _ { 0 } \right]$ and all $\theta \in [ - \pi , \pi ]$,
$$| h ( t , \theta ) | \leq e ^ { - \beta \left( \sigma _ { t } \theta \right) ^ { 2 } } \quad \text { or } \quad | h ( t , \theta ) | \leq e ^ { - \gamma \left( \sigma _ { t } | \theta | \right) ^ { 2 / 3 } }$$