Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. We denote $q = 1-p$ and $m = n^2$.
Let $i$ and $j$ be in $\{1, \ldots, n\}$. The smallest integer $k \geqslant 1$ such that the coefficient at row $i$, column $j$ of $M_k$ equals 1 is denoted $T_{i,j}$. Give the distribution of $T_{i,j}$.

Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. The smallest integer $k \geqslant 1$ such that the coefficient at row $i$, column $j$ of $M_k$ equals 1 is denoted $T_{i,j}$.
For an integer $k \geqslant 1$, give the value of $P(T_{i,j} \geqslant k)$.

Let $p \in ] 0,1 [$. What is the characteristic function of a random variable following a geometric distribution with parameter $p$ ?

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Show that for all $k \in \mathbb{N}^*$, the random variable $\nu_{p_k}(X) + 1$ follows the geometric distribution with parameter $\left(1 - p_k^{-s}\right)$.

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Deduce that there exists a real $K > 0$ independent of $p$ such that
$$\mathbf { E } \left( ( X - \mathbf { E } ( X ) ) ^ { 4 } \right) \leq \frac { K q } { p ^ { 4 } }$$

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.
The probability that $X = 3$ equals
(A) $\frac { 25 } { 216 }$
(B) $\frac { 25 } { 36 }$
(C) $\frac { 5 } { 36 }$
(D) $\frac { 125 } { 216 }$

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.
The probability that $X \geq 3$ equals
(A) $\frac { 125 } { 216 }$
(B) $\frac { 25 } { 36 }$
(C) $\frac { 5 } { 36 }$
(D) $\frac { 25 } { 216 }$

A department store is preparing many red envelopes for customers to draw during the Lunar New Year period, claiming that the activity will continue until all red envelopes are distributed. The drawing box contains 5 sticks, of which only 1 stick is marked ``Great Fortune'', and each stick has an equal chance of being drawn. Each customer draws one stick from the box, records it, puts it back, and draws again for the next round, drawing at most 3 times. When two consecutive draws result in ``Great Fortune'', the customer stops drawing and receives a red envelope. We can view whether each customer receives a red envelope as a Bernoulli trial. Let $X$ be the position of the first customer to receive a red envelope in the entire activity, and let $E(X)$ denote the expected value of the random variable $X$. Then $E(X) = $ \hspace{2cm}. (Round to the nearest integer)