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 $S$.

Given is a Bernoulli chain with length $n$ and success probability $p$. Explain that for all $k \in \{ 0 ; 1 ; 2 ; \ldots ; n \}$ the relationship $B ( n ; p ; k ) = B ( n ; 1 - p ; n - k )$ holds.
A company organizes trips with an excursion ship that has space for 60 passengers.
(1) [3 marks] Consider a trip where the ship is fully booked. Among the passengers are adults, teenagers and children. Half of the passengers eat ice cream during the trip, of the adults only one in three, of the teenagers and children 75\%. Calculate how many adults participate in the trip.
To participate in a trip, one must make a reservation in advance without having to pay the fare yet. Based on experience, some of the people with reservations do not appear for the trip. For the 60 available seats, the company therefore allows up to 64 reservations. It should be assumed that 64 reservations are actually made for each trip. If more than 60 people with reservations appear for the trip, only 60 of them can participate; the rest must be turned away. The random variable $X$ describes the number of people with reservations who do not appear for the trip. For simplicity, it should be assumed that $X$ is binomially distributed, where the probability that a randomly selected person with a reservation does not appear for the trip is 10\%. The table shown supplements the approved reference material.
Binomial distribution cumulative; $k \mapsto \sum _ { i = 0 } ^ { k } B ( n ; p ; i )$

n	k	$\mathrm { p } = 0.10$	$\mathrm { p } = 0.11$	$\mathrm { p } = 0.12$	$\mathrm { p } = 0.13$	$\mathrm { p } = 0.14$	$\mathrm { p } = 0.15$	$\mathrm { p } = 0.16$	p = 0.17
\multirow[t]{7}{*}{64}	0	0.00118	0.00058	0.00028	0.00013	0.00006	0.00003	0.00001	0.00001
	1	0.00956	0.00514	0.00272	0.00142	0.00073	0.00037	0.00019	0.00009
	2	0.03891	0.02290	0.01321	0.00748	0.00417	0.00228	0.00123	0.00065
	3	0.10629	0.06827	0.04277	0.02620	0.01572	0.00924	0.00533	0.00302
	4	0.22047	0.15377	0.10425	0.06886	0.04439	0.02797	0.01725	0.01043
	5	0.37271	0.28059	0.20485	0.14534	0.10040	0.06763	0.04450	0.02863
	...	...	...	...	...	...	...	...	...

(2a) [1 marks] Give a reason why the assumption that the random variable $X$ is binomially distributed is a simplification in the context of the problem.
(2b) [3 marks] Determine the probability that no person with a reservation needs to be turned away.
(2c) [3 marks] For the company, it would be helpful if the probability of having to turn away at least one person with a reservation were at most one percent. For this, the probability that a randomly selected person with a reservation does not appear for the trip would need to be at least a certain value. Determine this value to the nearest whole percent.
The company sets up an online portal for reservations and suspects that this could increase the proportion of people with reservations who do not appear for the respective trip. As a basis for deciding whether more than 64 reservations should be allowed per trip in the future, the null hypothesis "The probability that a randomly selected person with a reservation does not appear for the trip is at most 10\%." is to be tested using a sample of 200 people with reservations at a significance level of 5\%. Before the test is conducted, it is determined that the number of possible reservations per trip will only be increased if the null hypothesis would need to be rejected based on the test result.
(2d) [5 marks] Determine the associated decision rule.
(2e) [3 marks] Decide whether the choice of the null hypothesis was primarily motivated by the interest in having fewer empty seats or the interest in not having to turn away more people with reservations. Justify your decision.
(2f) [2 marks] Describe the associated Type II error and the resulting consequence in the context of the problem.

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 $1/2$.

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits, and $X$ follows the distribution determined in Q16. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ Using Markov's inequality, prove that if $r \leqslant 2 \frac{1-\alpha}{1-p}$, then condition (II.2) is satisfied.

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.

Problem 6
Company A owns multiple factories $i ( i = 1,2 , \cdots )$. Suppose that the probability of producing defective goods in a factory $i$ is $P _ { i }$, and that $N _ { i }$ goods are randomly sampled and shipped from the factory. Here, $P _ { i }$ is sufficiently small, and each factory does not affect any other.
I. Show the probability $f ( i , k )$, which is the probability of $k$ defective goods existing within $N _ { i }$ goods shipped from a factory $i$. Here, $k$ is a non-negative integer.
II. Show that $f ( i , k ) \rightarrow \frac { e ^ { - \lambda _ { i } } \lambda _ { i } ^ { k } } { k ! }$ when $N _ { i } \rightarrow \infty$. Here, when calculating the limit of $f ( i , k ) , \lambda _ { i }$ is a constant, where $\lambda _ { i } = N _ { i } P _ { i }$.
In the following questions, assume that $f ( i , k ) = \frac { e ^ { - \lambda _ { i } } \lambda _ { i } ^ { k } } { k ! }$.
III. Suppose that goods are shipped from two factories as shown in Table 1. Find the probability of two defective goods being contained within all shipped goods.

\begin{tabular}{ c } Factory number

$( i )$

&

Probability of defectiveness

$\left( P _ { i } \right)$

&

Number of shipped goods

$\left( N _ { i } \right)$

\hline 1 & 0.01 & 500 \hline 2 & 0.02 & 300 \hline \end{tabular}
IV. Find the probability of $k$ defective goods being contained within all shipped goods under the same conditions as in Question III.
V. Suppose that $P _ { i } = 0.001 i$ in five factories $i ( i = 1,2,3,4,5 )$ and the same number ($N _ { c}$) of goods are shipped from all these factories.
Find the maximum value of $N _ { c }$ for which the expected number of defective goods out of all shipped goods is equal to or less than 3.