18.
(1)
The probability distribution of $X$ is:
$$\begin{gathered} P ( X = 0 ) = 0.2 \\ P ( X = 20 ) = 0.8 \times ( 1 - 0.6 ) = 0.32 \\ P ( X = 100 ) = 0.8 \times 0.6 = 0.48 \end{gathered}$$
(2)
Given that type A questions are answered first, the mathematical expectation is...

Given that the premium for a certain insurance is 0.4 ten thousand yuan. For the first 3 claims, each claim pays 0.8 ten thousand yuan; the 4th claim pays 0.6 ten thousand yuan.

Number of Claims	0	1	2	3	4
Number of Policies	800	100	60	30	10

A sample of 100 policies is drawn from the population. Using frequency to estimate probability:
(1) Find the probability that a randomly selected policy has at least 2 claims;
(2) (i) Gross profit is the difference between premium and claim amount. Let gross profit be $X$. Estimate the mathematical expectation of $X$;
(ii) If policies with no claims have their premium reduced by 4\% in the next insurance period, and policies with claims have their premium increased by 20\%, estimate the mathematical expectation of gross profit for the next insurance period.

A shooting competition is divided into two stages. Each participating team consists of two members. The specific rules are as follows: In the first stage, one team member shoots 3 times. If all 3 shots miss, the team is eliminated with a score of 0. If at least one shot is made, the team advances to the second stage, where the other team member shoots 3 times, earning 5 points for each made shot and 0 points for each missed shot. The team's final score is the total points from the second stage. A participating team consists of members A and B. Let the probability that A makes each shot be $p$, and the probability that B makes each shot be $q$. Each shot is independent.
(1) If $p = 0.4$ and $q = 0.5$, with A participating in the first stage, find the probability that the team's score is at least 5 points.
(2) Assume $0 < p < q$.
(i) To maximize the probability that the team's score is 15 points, who should participate in the first stage?
(ii) To maximize the expected value of the team's score, who should participate in the first stage?

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables, defined on a probability space $(\Omega, \mathcal{A}, P)$ and following the same Bernoulli distribution with parameter $p$.
Calculate the probability that $X_1, \ldots, X_n$ are all equal.

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables, defined on a probability space $(\Omega, \mathcal{A}, P)$ and following the same Bernoulli distribution with parameter $p$.
Let $i$ and $j$ be in $\{1, \ldots, n\}$. Give the distribution of the random variable $X_{i,j} = X_i \times X_j$.

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables following the same Bernoulli distribution with parameter $p$. Let $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
Give the distribution, expectation and variance of the random variables $\operatorname{tr}(M)$ and $\operatorname{rg}(M)$.

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables following the same Bernoulli distribution with parameter $p$. Let $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
What is the probability that $M$ has two distinct eigenvalues?

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. We consider a function $h$ uniformly continuous and bounded on $\mathbb{R}$ such that for all $x \in \mathbb{R}$, $h(x) \leqslant h(0)$ and $$h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$$ We recall that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$. Show that for all $n \in \mathbb{N}$ and $x \geqslant 0$ such that $\mathbb{P}\left(S_n = x\right) > 0$, we have $h(-x) = h(0)$.

Let $x \in \mathbb{R}$ such that $x > 1$. Show that we define the probability distribution of a random variable $X$ taking values in $\mathbb{N}^{*}$ by setting $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$

Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Let $(q_{1}, \ldots, q_{n}) \in \mathcal{P}^{n}$ be an $n$-tuple of distinct prime numbers. Show that the events $(X \in q_{1}\mathbb{N}^{*}), \ldots, (X \in q_{n}\mathbb{N}^{*})$ are mutually independent.

Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ For all $n \in \mathbb{N}^{*}$, denote by $B_{n}$ the event $B_{n} = \bigcap_{k=1}^{n} (X \notin p_{k}\mathbb{N}^{*})$, where $p_1 < p_2 < \cdots$ are the prime numbers in increasing order.
Show that $\lim_{n \rightarrow \infty} \mathbb{P}(B_{n}) = \mathbb{P}(X = 1)$. Deduce that $$\forall x \in {]1,+\infty[}, \quad \frac{1}{\zeta(x)} = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} \left(1 - \frac{1}{p_{k}^{x}}\right)$$

Let $x \in \mathbb{R}$ such that $x > 1$. Let $X$ and $Y$ be two independent random variables each following a zeta probability distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \mathbb{P}(Y = n) = \frac{1}{\zeta(x) n^{x}}$$ Let $A$ be the event ``No prime number divides $X$ and $Y$ simultaneously''. For all $n \in \mathbb{N}^{*}$, denote by $C_{n}$ the event $$C_{n} = \bigcap_{k=1}^{n} \left((X \notin p_{k}\mathbb{N}^{*}) \cup (Y \notin p_{k}\mathbb{N}^{*})\right)$$ Express the event $A$ using the events $C_{n}$. Deduce that $$\mathbb{P}(A) = \frac{1}{\zeta(2x)}$$

Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$. We denote by $W_{n} = U_{n} \wedge V_{n}$ (the GCD of $U_n$ and $V_n$).
For all $k \in \mathbb{N}^{*}$, show that $$\mathbb{P}\left(W_{n} \in k\mathbb{N}^{*}\right) = \left(\frac{\lfloor n/k \rfloor}{n}\right)^{2}$$

Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$, and $W_{n} = U_{n} \wedge V_{n}$. We admit that, for all $k \in \mathbb{N}^{*}$, the sequence $(\mathbb{P}(W_{n} = k))_{n \in \mathbb{N}^{*}}$ converges to a real number $\ell_{k}$.
Show that $$\forall \varepsilon > 0, \quad \exists M \in \mathbb{N}^{*} \text{ such that } \forall m \in \mathbb{N}^{*},\ m \geqslant M \Longrightarrow 1 - \varepsilon \leqslant \sum_{k=1}^{m} \ell_{k} \leqslant 1$$

Let $(\ell_k)_{k \in \mathbb{N}^*}$ be the limits defined in Q33, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.
Using the result of Q33, deduce that $(\ell_{k})_{k \in \mathbb{N}^{*}}$ defines a probability distribution on $\mathbb{N}^{*}$.

Let $W$ be a random variable on $\mathbb{N}^{*}$ that follows the probability distribution $(\ell_k)_{k \in \mathbb{N}^*}$, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.
We admit that for all $B \subseteq \mathbb{N}^*$, $\mathbb{P}(W \in B) = \lim_{n \to \infty} \mathbb{P}(W_n \in B)$, and that if $X$ and $Y$ are two random variables taking values in $\mathbb{N}^*$ with $\mathbb{P}(X \in a\mathbb{N}^*) = \mathbb{P}(Y \in a\mathbb{N}^*)$ for all $a \in \mathbb{N}^*$, then $X$ and $Y$ have the same distribution.
Specify the distribution of $W$. By considering $\ell_{1}$, what can we then conclude?

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables. We denote, for every integer $n \geqslant 1$, $S_{n} = \sum_{j=1}^{n} X_{j}$. Let $k$ be an integer such that $-n \leqslant k \leqslant n$. Show that, if $n$ and $k$ do not have the same parity, then $\mathbb{P}\left(S_{n} = k\right) = 0$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables. We denote $S_{n} = \sum_{j=1}^{n} X_{j}$. Let $k$ be an integer such that $-n \leqslant k \leqslant n$. Show that, if $n$ and $k$ have the same parity, $\mathbb{P}\left(S_{n} = k\right) = \binom{n}{(k+n)/2} \frac{1}{2^{n}}$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For $x$ real, $\lfloor x \rfloor$ denotes the integer part of $x$. For all real numbers $\delta > 0$ and $\tau > 0$, calculate $\mathbb{V}\left(\delta S_{\lfloor 1/\tau \rfloor}\right)$, the variance of the random variable $\delta S_{\lfloor 1/\tau \rfloor}$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. Show that, for every real number $\delta$, $\mathbb{V}\left(\delta S_{\lfloor 1/\tau \rfloor}\right)$ is equivalent to $\frac{\delta^{2}}{\tau}$, as $\tau$ tends to 0 from above.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For every $n \in \mathbb{N}^{*}$ and every $k \in \mathbb{Z}$, by setting $p_{n}(k) = \mathbb{P}\left(S_{n} = k\right)$, show that $$\frac{p_{n+1}(k) - p_{n}(k)}{\tau} = \frac{\delta^{2}}{2\tau} \frac{p_{n}(k+1) - 2p_{n}(k) + p_{n}(k-1)}{\delta^{2}}$$

We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:

• we randomly draw a ball from the urn;
• we replace the drawn ball in the urn;
• we add to the urn a ball of the same color as the drawn ball.

We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws.
Let $n$ and $k$ be two integers greater than or equal to 1. Establish that $$P(X_{n} = k) = \frac{k-1}{n+1} P(X_{n-1} = k-1) + \frac{n+1-k}{n+1} P(X_{n-1} = k).$$

We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:

• we randomly draw a ball from the urn;
• we replace the drawn ball in the urn;
• we add to the urn a ball of the same color as the drawn ball.

We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws.
Identify the distribution of $X_{n}$ and give its expectation.

We consider a balanced urn with $a_{0} = 1, b_{0} = 0, a = d = 0$ and $b = c = 1$. In other words, the urn initially contains one white ball and, at each draw, we add a ball of the opposite color to the one that was drawn.
By listing all possible outcomes, give the distribution of $X_{3}$.

We consider a balanced urn with $a_{0} = 1, b_{0} = 0, a = d = 0$ and $b = c = 1$. In other words, the urn initially contains one white ball and, at each draw, we add a ball of the opposite color to the one that was drawn. For all real $u$ and $v$, we set $P_{0}(u,v) = u^{a_{0}} v^{b_{0}}$ and $P_{n}(u,v) = \sum_{\omega \in \Omega_{n}} u^{b(\omega)} v^{n(\omega)}$.
Verify that $P_{3}(u,v) = uv^{3} + 4u^{2}v^{2} + u^{3}v$.