Exercise 3

The ``base64'' encoding, used in computing, allows messages and other data such as images to be represented and transmitted using 64 characters: the 26 uppercase letters, the 26 lowercase letters, the digits 0 to 9 and two other special characters. Parts A, B and C are independent.

Part A

In this part, we are interested in sequences of 4 characters in base64. For example, ``gP3g'' is such a sequence. In a sequence, order must be taken into account: the sequences ``m5C2'' and ``5C2m'' are not identical.

1. Determine the number of possible sequences.
2. Determine the number of sequences if we require that the 4 characters are pairwise different.
3. a. Determine the number of sequences containing no uppercase letter A. b. Deduce the number of sequences containing at least one uppercase letter A. c. Determine the number of sequences containing exactly one uppercase letter A. d. Determine the number of sequences containing exactly two uppercase letters A.

Part B

We are interested in the transmission of a sequence of 250 characters from one computer to another. We assume that the probability that a character is incorrectly transmitted is equal to 0.01 and that the transmissions of the different characters are independent of each other. We denote by $X$ the random variable equal to the number of characters incorrectly transmitted.

1. We admit that the random variable $X$ follows the binomial distribution. Give its parameters.
2. Determine the probability that all characters are correctly transmitted. The exact expression will be given, then an approximate value to $10^{-3}$ near.
3. What do you think of the following statement: ``The probability that more than 16 characters are incorrectly transmitted is negligible''?

Part C

We are now interested in the transmission of 4 sequences of 250 characters. We denote by $X_1, X_2, X_3$ and $X_4$ the random variables corresponding to the numbers of characters incorrectly transmitted during the transmission of each of the 4 sequences. We admit that the random variables $X_1, X_2, X_3$ and $X_4$ are independent of each other and follow the same distribution as the random variable $X$ defined in part B. We denote by $S = X_1 + X_2 + X_3 + X_4$. Determine, by justifying, the expectation and the variance of the random variable $S$.

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 $p$.
What is the distribution of $S = X_1 + \ldots + X_n$? A proof of the stated result is expected.

In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. We have shown that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}\right| \geqslant \varepsilon\right) \leqslant 2 \exp\left(-n \frac{\varepsilon^{2}}{2c^{2}}\right)$.
Let $n$ be a non-zero natural number, $p$ an element of the interval $]0,1[$ and $Z$ a random variable following a binomial distribution with parameter $(n, p)$. Using the previous question, bound $\mathbb{P}\left(\left|\frac{Z}{n}-p\right| \geqslant \varepsilon\right)$ as a function of $n, p$ and $\varepsilon$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables (taking values in $\{1,-1\}$ each with probability $1/2$). For every $n \in \mathbb{N}^{*}$, we set $Y_{n} = \frac{1}{2}\left(X_{n}+1\right)$ and $Z_{n} = \sum_{j=1}^{n} Y_{j}$. Determine the distribution of the random variable $Y_{n}$ and that of the random variable $Z_{n}$.

Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. We assume that $C$ is a closed convex set of $E$ that meets $X(\Omega)$ in a single vector $u$. Show that $\frac{1}{4}d(X, u)^{2}$ follows a binomial distribution with parameters $n$ and $1/2$.

Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. We assume that $C$ is a closed convex set of $E$ that meets $X(\Omega)$ in a single vector $u$. Show that $\frac{1}{4} d(X, u)^{2}$ follows a binomial distribution with parameters $n$ and $1/2$.

Establish the following identity: for $( s , i , r ) \in E$, for all $k \in \{ 0 , \cdots , s \}$,
$$\mathbf { P } \left( \Delta \tilde { S } _ { n } = - k \mid \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right) = \binom { s } { k } ( p ( i ) ) ^ { k } ( 1 - p ( i ) ) ^ { s - k }$$
where $p(i)$ is the probability for a susceptible person to be infected during the day (as found in question 21), and the $s$ susceptible persons act independently.