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?

Let $t$ be a strictly positive real number. Using questions 20 and 12, and the result that if $u$ and $v$ are functions from $\mathbb{R}$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}$ and satisfying $\mathcal{F}(u) = \mathcal{F}(v)$, then $u = v$, deduce the existence of a real $\lambda_{t,\sigma}$ such that $$f(t, \cdot) = \lambda_{t,\sigma} g_{\sqrt{\sigma^{2}+2t}}$$

Deduce that, for any strictly positive real $t$, $f(t, \cdot) = g_{\sqrt{\sigma^{2}+2t}}$.

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}.$$
Justify $$\forall n \in \mathbb{N}^{\star}, \quad \mathbb{P}(Y_n \in [0,1[) = 1.$$

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 $$\forall n \in \mathbb{N}^{\star}, \forall x \in D_n, \quad G_n(x) = x.$$

In the general model of a Pólya urn ($b = c = 0$, $a = d$), using the results established so far (in particular that $H = G$ on $D_{\rho}$), conclude that, for all integers $n$ and for all $k \in \llbracket 0, n \rrbracket$, $$P(X_{n} = a_{0} + ka) = \binom{n}{k} \frac{L_{k}(a_{0}/a) L_{n-k}(b_{0}/a)}{L_{n}(a_{0}/a + b_{0}/a)}$$

In the general model of a Pólya urn ($b = c = 0$, $a = d$), using the result of question 26, recover the result of question 10.

For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$ Determine the eigenvalues of $T$ and show that the associated eigenspaces are one-dimensional.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that if $p \neq \frac { 1 } { 2 }$, then $T$ admits an expectation.

We assume that $\Sigma_Y$ has $n$ distinct eigenvalues which we order in strictly decreasing order $\lambda_1 > \cdots > \lambda_n$. We equip ourselves with a vector $U_0$ such that $\mathbb{V}\left(U_0^\top Y\right) = \max_{U \in C} \mathbb{V}\left(U^\top Y\right)$, where $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. We denote $$C' = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \text{ and } U_0^\top U = 0 \right\}.$$
Determine the value of $\max_{U \in C'} \mathbb{V}\left(U^\top Y\right)$ and specify a vector $U_1 \in C'$ such that $$\max_{U \in C'} \mathbb{V}\left(U^\top Y\right) = \mathbb{V}\left(U_1^\top Y\right).$$

We assume that $\Sigma_Y$ has $n$ distinct eigenvalues which we order in strictly decreasing order $\lambda_1 > \cdots > \lambda_n$. We equip ourselves with a vector $U_0$ such that $\mathbb{V}\left(U_0^\top Y\right) = \max_{U \in C} \mathbb{V}\left(U^\top Y\right)$, and a vector $U_1 \in C'$ such that $\mathbb{V}\left(U_1^\top Y\right) = \max_{U \in C'} \mathbb{V}\left(U^\top Y\right)$, where $$C' = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \text{ and } U_0^\top U = 0 \right\}.$$
Calculate the covariance of the discrete random variables $U_0^\top Y$ and $U_1^\top Y$ (to simplify notation, one may assume $Y$ is centered, that is, $\mathbb{E}(Y) = 0$).

Consider a situation where products are produced sequentially. The events producing defective products are independent and identically distributed, and a defective product is produced with a probability of $\phi$ $(0 \leq \phi \leq 1)$. Suppose that the probability of producing a defective product follows the Beta distribution
$$\operatorname{Beta}_{\mathrm{a},\mathrm{b}}(x) = \frac{1}{\mathrm{B}(\mathrm{a},\mathrm{b})} x^{\mathrm{a}-1}(1-x)^{\mathrm{b}-1} \quad (0 \leq x \leq 1),$$
for real numbers $\mathrm{a}(>1)$ and $\mathrm{b}(>1)$. Note that the Beta function $\mathrm{B}(\mathrm{a},\mathrm{b})$ is defined as
$$\mathrm{B}(\mathrm{a},\mathrm{b}) = \int_0^1 t^{\mathrm{a}-1}(1-t)^{\mathrm{b}-1} \mathrm{d}t$$
In the Bayesian estimation, the parameter $\theta$ (in this case, $\phi$) that determines the probability is treated as the random variable and we assume that its distribution is described by $\pi(\theta)$. We calculate $\pi(\theta \mid A)$ by
$$\pi(\theta \mid A) = \frac{\pi(\theta) P(A \mid \theta)}{P(A)}$$
where $\pi(\theta \mid A)$ is the posterior probability, $P(A \mid \theta)$ is the conditional occurrence probability that the event $A$ is observed under $\theta$, and $\pi(\theta)$ is the prior probability.
We assume that $\phi$, the probability of producing a defective product, follows the prior probability $\operatorname{Beta}_{\mathrm{a},\mathrm{b}}(\phi)$. Let $Q(\boldsymbol{v} \mid \phi)$ be the conditional occurrence probability of $\boldsymbol{v}$ under $\phi$ and $Q_{\mathrm{a},\mathrm{b}}(\boldsymbol{v})$ be the occurrence probability of $\boldsymbol{v}$. Obtain the posterior probability after $\boldsymbol{v}$ occurs.

Consider a situation where products are produced sequentially. The events producing defective products are independent and identically distributed, and a defective product is produced with a probability of $\phi$ $(0 \leq \phi \leq 1)$. Suppose that the probability of producing a defective product follows the Beta distribution
$$\operatorname{Beta}_{\mathrm{a},\mathrm{b}}(x) = \frac{1}{\mathrm{B}(\mathrm{a},\mathrm{b})} x^{\mathrm{a}-1}(1-x)^{\mathrm{b}-1} \quad (0 \leq x \leq 1),$$
for real numbers $\mathrm{a}(>1)$ and $\mathrm{b}(>1)$. Note that the Beta function $\mathrm{B}(\mathrm{a},\mathrm{b})$ is defined as
$$\mathrm{B}(\mathrm{a},\mathrm{b}) = \int_0^1 t^{\mathrm{a}-1}(1-t)^{\mathrm{b}-1} \mathrm{d}t$$
By defining $v_i = 1$ if the $i$-th product is a defective product, and $v_i = 0$ if it is not defective, we get a series $\boldsymbol{v} = (v_1, \cdots, v_N)$, where the values can be 0 or 1. Let $N_d(\boldsymbol{v})$ be the number of observations with value of 1 in $\boldsymbol{v}$.
Suppose that $Q(\boldsymbol{v} \mid \phi)$ in Question II.2 is the occurrence probability obtained in Question II.1 and let $a = 2,\ b = 50$, obtain $Q_{2,50}(\boldsymbol{v})$.

Consider a situation where products are produced sequentially. The events producing defective products are independent and identically distributed, and a defective product is produced with a probability of $\phi$ $(0 \leq \phi \leq 1)$. Suppose that the probability of producing a defective product follows the Beta distribution
$$\operatorname{Beta}_{\mathrm{a},\mathrm{b}}(x) = \frac{1}{\mathrm{B}(\mathrm{a},\mathrm{b})} x^{\mathrm{a}-1}(1-x)^{\mathrm{b}-1} \quad (0 \leq x \leq 1),$$
for real numbers $\mathrm{a}(>1)$ and $\mathrm{b}(>1)$. Note that the Beta function $\mathrm{B}(\mathrm{a},\mathrm{b})$ is defined as
$$\mathrm{B}(\mathrm{a},\mathrm{b}) = \int_0^1 t^{\mathrm{a}-1}(1-t)^{\mathrm{b}-1} \mathrm{d}t$$
In Question II.3, with $a=2,\ b=50$, show that the posterior probability becomes the Beta distribution $\operatorname{Beta}_{\mathrm{a}^{\prime},\mathrm{b}^{\prime}}(\phi)$, and obtain $\mathrm{a}^{\prime}$ and $\mathrm{b}^{\prime}$.

Consider a situation where products are produced sequentially. The events producing defective products are independent and identically distributed, and a defective product is produced with a probability of $\phi$ $(0 \leq \phi \leq 1)$. Suppose that the probability of producing a defective product follows the Beta distribution
$$\operatorname{Beta}_{\mathrm{a},\mathrm{b}}(x) = \frac{1}{\mathrm{B}(\mathrm{a},\mathrm{b})} x^{\mathrm{a}-1}(1-x)^{\mathrm{b}-1} \quad (0 \leq x \leq 1),$$
for real numbers $\mathrm{a}(>1)$ and $\mathrm{b}(>1)$. Note that the Beta function $\mathrm{B}(\mathrm{a},\mathrm{b})$ is defined as
$$\mathrm{B}(\mathrm{a},\mathrm{b}) = \int_0^1 t^{\mathrm{a}-1}(1-t)^{\mathrm{b}-1} \mathrm{d}t$$
In Question II.4, where the posterior probability is the Beta distribution $\operatorname{Beta}_{\mathrm{a}^{\prime},\mathrm{b}^{\prime}}(\phi)$ with $a=2,\ b=50$, obtain $\phi$ that gives the maximum likelihood estimate (that maximizes the posterior probability).

Consider a region $R$ defined by $0 < x < 1$ and $0 < y < 1$ in the $x y$-plane. We randomly select a point on $R$ and refer to the selected point as A. We assume that A is uniformly distributed on $R$. Let AB be a perpendicular line from A to the $y$-axis and AC be a perpendicular line from $A$ to the $x$-axis as shown in the figure. We call rectangle $OCAB$ as "the rectangle of A", where O denotes the origin. Let $S$ be a random variable representing the area of the rectangle of A. Answer the following questions.
(1) Calculate the expectation value of $S$.
(2) Calculate the probability that $S \leq r$ holds, where $0 < r < 1$.
(3) Calculate the probability density function of $S$.
Again consider the region $R$. Let $n$ be a positive integer. We select $n$ points on $R$ and refer to the selected points as $\mathrm{A}_1, \mathrm{~A}_2, \ldots, \mathrm{~A}_n$. We assume that each of the points is uniformly distributed on $R$, and $\mathrm{A}_i$ and $\mathrm{A}_j$ for $i \neq j$ are selected independently. Answer the following question.
(4) Let $S_i$ be a random variable representing the area of the rectangle of $\mathrm{A}_i$. Let $Z$ be a random variable which is the minimum of $S_1, S_2, \ldots, S_n$. Calculate the probability density function of $Z$.