2. Let $Y$ denote the random variable which, for a small jar randomly selected from the production of line $\mathrm { F } _ { 2 }$, associates its sugar content. We assume that $Y$ follows the normal distribution with mean $m _ { 2 } = 0.17$ and standard deviation $\sigma _ { 2 }$. We further assume that the probability that a small jar randomly selected from the production of line $\mathrm { F } _ { 2 }$ is compliant equals 0.99 . Let Z be the random variable defined by $Z = \frac { Y - m _ { 2 } } { \sigma _ { 2 } }$. a. What distribution does the random variable $Z$ follow? b. Determine, as a function of $\sigma _ { 2 }$, the interval to which $Z$ belongs when $Y$ belongs to the interval $[ 0.16 ; 0.18 ]$. c. Deduce an approximate value to $10 ^ { - 3 }$ near of $\sigma _ { 2 }$.
You may use the table given below, in which the random variable $Z$ follows the normal distribution with mean 0 and standard deviation 1 .

$\beta$	$P ( - \beta \leqslant Z \leqslant \beta )$
2.4324	0.985
2.4573	0.986
2.4838	0.987
2.5121	0.988
2.5427	0.989
2.5758	0.990
2.6121	0.991
2.6521	0.992
2.6968	0.993

Exercise 3

Common to all candidates

Given a real number $k$, we consider the function $f _ { k }$ defined on $\mathbb { R }$ by
$$f _ { k } ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - k x } }$$
The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ).

Part A

In this part we choose $k = 1$. We have therefore, for all real $x , f _ { 1 } ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - x } }$. The graph $\mathscr { C } _ { 1 }$ of the function $f _ { 1 }$ in the coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ) is given in APPENDIX, to be returned with your answer sheet.

1. Determine the limits of $f _ { 1 } ( x )$ as $x \to + \infty$ and as $x \to - \infty$ and give a graphical interpretation of the results obtained.
2. Prove that, for all real $x , f _ { 1 } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$.
3. Let $f _ { 1 } ^ { \prime }$ denote the derivative function of $f _ { 1 }$ on $\mathbb { R }$. Calculate, for all real $x , f _ { 1 } ^ { \prime } ( x )$.

Deduce the variations of the function $f _ { 1 }$ on $\mathbb { R }$.

Exercise 1 (5 points)

A large cosmetics brand launches a new moisturizing cream.

Part A: Packaging of jars

This brand wishes to sell the new cream in a 50 mL package and has jars with a maximum capacity of 55 mL for this purpose.
A jar of cream is said to be non-compliant if it contains less than 49 mL of cream.

1. Several series of tests lead to modeling the quantity of cream, expressed in mL, contained in each jar by a random variable $X$ which follows the normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 1.2$. Calculate the probability that a jar of cream is non-compliant.
2. The proportion of non-compliant jars of cream is judged to be too large. By modifying the viscosity of the cream, we can change the value of the standard deviation of the random variable $X$, without modifying its mean $\mu = 50$. We want to reduce to 0.06 the probability that a randomly chosen jar is non-compliant. We denote $\sigma ^ { \prime }$ the new standard deviation, and $Z$ the random variable equal to $\frac { X - 50 } { \sigma ^ { \prime } }$ a. Specify the distribution followed by the random variable $Z$. b. Determine an approximate value of the real number $u$ such that $p ( Z \leqslant u ) = 0.06$. c. Deduce the expected value of $\sigma ^ { \prime }$.
3. A shop orders 50 jars of this new cream from its supplier.

We consider that the work on the viscosity of the cream has made it possible to achieve the set objective and therefore that the proportion of non-compliant jars in the sample is 0.06. Let $Y$ be the random variable equal to the number of non-compliant jars among the 50 jars received. a. We admit that $Y$ follows a binomial distribution. Give its parameters. b. Calculate the probability that the shop receives two non-compliant jars or fewer than two non-compliant jars.

Part B: Advertising campaign

A consumer association decides to estimate the proportion of people satisfied by the use of this cream. It conducts a survey among people using this product. Out of 140 people interviewed, 99 declare themselves satisfied. Estimate, by confidence interval at the 95\% threshold, the proportion of satisfied people among the users of the cream.

A football is compliant with regulations if it meets, depending on its size, two conditions simultaneously (on its mass and on its circumference). In particular, a standard-sized football is compliant with regulations when its mass, expressed in grams, belongs to the interval [410;450] and its circumference, expressed in centimetres, belongs to the interval [68;70].

1. Let $X$ denote the random variable which, for each standard-sized football chosen at random in the company, associates its mass in grams. It is admitted that $X$ follows a normal distribution with mean 430 and standard deviation 10. Determine an approximate value to $10 ^ { - 3 }$ of the probability $P ( 410 \leqslant X \leqslant 450 )$.
2. Let $Y$ denote the random variable which, for each standard-sized football chosen at random in the company, associates its circumference in centimetres. It is admitted that $Y$ follows a normal distribution with mean 69 and standard deviation $\sigma$. Determine the value of $\sigma$, to the nearest hundredth, knowing that $97 \%$ of standard-sized footballs have a circumference compliant with regulations. You may use the following result: when $Z$ is a random variable that follows the standard normal distribution, then $P ( - \beta \leqslant Z \leqslant \beta ) = 0,97$ for $\beta \approx 2,17$.

Each cone is filled with vanilla ice cream. We denote by $Y$ the random variable which, to each cone, associates the mass (expressed in grams) of ice cream it contains. It is assumed that $Y$ follows a normal distribution $\mathscr{N}\left(110 ; \sigma^{2}\right)$, with mean $\mu = 110$ and standard deviation $\sigma$.
An ice cream is considered marketable when the mass of ice cream it contains belongs to the interval $[104; 116]$.
Determine an approximate value to $10^{-1}$ of the parameter $\sigma$ such that the probability of the event ``the ice cream is marketable'' is equal to 0.98.

A factory produces mineral water in bottles. When the calcium level in a bottle is less than $6.5 \mathrm { mg }$ per litre, the water in that bottle is said to be very low in calcium.
Let $X$ be the random variable that, for each bottle randomly selected from the daily production of source A, associates the calcium level of the water it contains. We assume that $X$ follows a normal distribution with mean 8 and standard deviation 1.6. Let $Y$ be the random variable that, for each bottle randomly selected from the daily production of source B, associates the calcium level it contains. We assume that $Y$ follows a normal distribution with mean 9 and standard deviation $\sigma$.

1. Determine the probability that the calcium level measured in a bottle randomly taken from the daily production of source A is between $6.4 \mathrm { mg }$ and $9.6 \mathrm { mg }$.
2. Calculate the probability $p ( X \leqslant 6.5 )$.
3. Determine $\sigma$ knowing that the probability that a bottle randomly selected from the daily production of source B contains water that is very low in calcium is 0.1.

Exercise 5 — Candidates who have not followed the specialization course In this exercise, all requested probabilities will be rounded to $10 ^ { - 4 }$. We study a model of automobile air conditioner composed of a mechanical module and an electronic module. If a module fails, it is replaced.
Part A: Study of mechanical module failures An automobile maintenance company has found, through a statistical study, that the operating time (in months) of the mechanical module can be modeled by a random variable $D$ that follows a normal distribution with mean $\mu = 50$ and standard deviation $\sigma$:

1. Determine the rounding to $10 ^ { - 4 }$ of $\sigma$ knowing that the statistical service indicates that $P ( D \geqslant 48 ) = 0,7977$.

For the rest of this exercise, we will take $\sigma = 2,4$.

1. Determine the probability that the operating time of the mechanical module is between 45 and 52 months.
2. Determine the probability that the mechanical module of an air conditioner that has been operating for 48 months will continue to function for at least 6 more months.

Part B: Study of electronic module failures On the same air conditioner model, the automobile maintenance company has found that the operating time (in months) of the electronic module can be modeled by a random variable $T$ that follows an exponential distribution with parameter $\lambda$.

1. Determine the exact value of $\lambda$, knowing that the statistical service indicates that $P ( 0 \leqslant T \leqslant 24 ) = 0,03$.

For the rest of this exercise, we will take $\boldsymbol { \lambda } = 0,00127$.

1. Determine the probability that the operating time of the electronic module is between 24 and 48 months.
2. a. Prove that, for all positive real numbers $t$ and $h$, we have: $P _ { T \geqslant t } ( T \geqslant t + h ) = P ( T \geqslant h )$, that is, the random variable $T$ is memoryless. b. The electronic module of the air conditioner has been operating for 36 months. Determine the probability that it will continue to function for the next 12 months.

Part C: Mechanical and electronic failures We admit that the events ( $D \geqslant 48$ ) and ( $T \geqslant 48$ ) are independent. Determine the probability that the air conditioner does not fail before 48 months.
Part D: Special case of a company garage
A garage of the company has studied the maintenance records of 300 air conditioners over 4 years old. It finds that 246 of them have their mechanical module in working order for 4 years. Should this report call into question the result given by the company's statistical service, namely that $P ( D \geqslant 48 ) = 0,7977$? Justify the answer.

The company ``Bonne Mamie'' uses a machine to fill jam jars on a production line. We denote by $X$ the random variable that associates to each jar of jam produced the mass of jam it contains, expressed in grams. In the case where the machine is correctly adjusted, we admit that $X$ follows a normal distribution with mean $\mu = 125$ and standard deviation $\sigma$.

1. a. For any positive real number $t$, determine a relationship between $$P ( X \leqslant 125 - t ) \text { and } P ( X \geqslant 125 + t ) .$$ b. We know that $2.3\%$ of the jam jars contain less than 121 grams of jam. Using the previous relationship, determine $$P ( 121 \leqslant X \leqslant 129 ) .$$
2. Determine a value rounded to the nearest unit of $\sigma$ such that $$P ( 123 \leqslant X \leqslant 127 ) = 0.68 .$$

In the rest of the exercise, we assume that $\boldsymbol { \sigma } = \mathbf { 2 }$.

1. We estimate that a jar of jam is compliant when the mass of jam it contains is between 120 and 130 grams. a. We randomly choose a jar of jam from the production. Determine the probability that this jar is compliant. The result will be given rounded to $10 ^ { - 4 }$. b. We randomly choose a jar from those with a jam mass less than 130 grams. What is the probability that this jar is not compliant? The result will be given rounded to $10 ^ { - 4 }$.
2. We admit that the probability, rounded to $10 ^ { - 3 }$, that a jar of jam is compliant is 0.988. We randomly choose 900 jars from the production. We observe that 871 of these jars are compliant. At the 95\% threshold, can we reject the following hypothesis: ``The machine is correctly adjusted''?

The mass in grams of melons from market gardener B is modelled by a random variable $M_\mathrm{B}$ that follows a normal distribution with mean 1050 and unknown standard deviation $\sigma$. Melons are described as ``compliant'' if their mass is between 900 g and 1200 g. The retailer observes that $85\%$ of melons supplied by market gardener B are compliant. Determine the standard deviation $\sigma$ of the random variable $M_\mathrm{B}$. Give the value rounded to the nearest integer.

Part of the city's DVD stock consists of animated films intended for young audiences. An animated film is chosen at random and we denote by $X$ the random variable that gives the duration, in minutes, of this film. $X$ follows a normal distribution with mean $\mu = 80$ min and standard deviation $\sigma$. Furthermore, it is estimated that $P(X \geqslant 92) = 0{,}10$.

1. Determine the real number $\sigma$ and give an approximate value to 0.01.
2. A child watches an animated film whose duration he does not know. Knowing that he has already watched one hour and thirty minutes, what is the probability that the film ends within the next five minutes?

The standard inner diameter of a bearing on a roller wheel is 8 mm. Let $X$ denote the random variable giving in mm the diameter of a bearing and we assume that $X$ follows a normal distribution with mean 8 and standard deviation 0.1.
A bearing is said to be compliant if its diameter is between $7.8 \mathrm{~mm}$ and $8.2 \mathrm{~mm}$.

1. Calculate the probability that a bearing is compliant.
2. Supplier $B$ sells its bearings in batches of 16 and claims that only $5\%$ of its bearings are non-compliant. The club president, who bought 30 batches from him, finds that 38 bearings are non-compliant. Does this check call into question supplier B's claim? An asymptotic fluctuation interval at the $95\%$ threshold may be used.
3. The bearing manufacturer of this supplier decides to improve the production of its bearings. The adjustment of the machine that manufactures them is modified so that $96\%$ of the bearings are compliant. We assume that after adjustment the random variable $X$ follows a normal distribution with mean 8 and standard deviation $\sigma$. a. What is the distribution followed by $\frac{X - 8}{\sigma}$? b. Determine $\sigma$ so that the manufactured bearing is compliant with a probability equal to 0.96.

Part B

The travel time for Louise, in minutes, between her home and work, can be modeled by a random variable $X$ that follows a normal distribution with mean 28 and standard deviation 5.

1. Calculate $P ( X \leqslant 25 )$.
2. Calculate the probability that the travel time is between 18 and 38 minutes.
3. Determine the travel duration $d$, rounded to the nearest minute, such that $P ( X \geqslant d ) = 0.1$.
4. Louise has now found a faster route. From now on, the travel time, in minutes, can be modeled by a random variable $Y$ that follows a normal distribution with mean 26 and standard deviation $\sigma$. We know that $P ( Y \geqslant 30 ) = 0.1$. Determine $\sigma$ rounded to the nearest hundredth.

A machine manufactures balls intended for a game of chance. The mass in grams of each of these balls can be modeled by a random variable $M$ following a normal distribution with mean 52 and standard deviation $\sigma$. Balls whose mass is between 51 and 53 grams are said to be compliant.

1. With the initial settings of the machine we have $\sigma = 0.437$. Under these conditions, calculate the probability that a ball manufactured by this machine is compliant. An approximate value to $10 ^ { - 1 }$ near the result will be given.
2. It is considered that the machine is correctly adjusted if at least $99 \%$ of the balls it manufactures are compliant. Determine an approximate value of the largest value of $\sigma$ that allows us to affirm that the machine is correctly adjusted.