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 $t$, $f(t, \cdot) = g_{\sqrt{\sigma^{2}+2t}}$.

We consider $g(X)$ where $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ is a random variable with independent Rademacher coefficients and $g(M) = \|M \cdot u\|$ for a fixed unit vector $u$. Justify that $g(X)$ admits at least one median. One may consider the function $G$ from $\mathbb{R}$ to $\mathbb{R}$ such that, for every real number $t$, $G(t) = \mathbb{P}(g(X) \leqslant t)$, and examine the set $G^{-1}([1/2, 1])$.

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

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.

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ 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 $$\forall n \in \mathbb{N}^{\star}, \quad \mathbb{P}(Y_n \in [0,1[) = 1.$$

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.

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ 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 $Z$.