Exercise 1 — Part B
We randomly choose a person who came to the multisports centre on a weekend. We denote $T_1$ the random variable giving their total waiting time in minutes before access to sports activities during Saturday and $T_2$ the random variable giving their total waiting time in minutes before access to sports activities during Sunday. We admit that:

• $T_1$ follows a probability distribution with expectation $E(T_1) = 40$ and standard deviation $\sigma(T_1) = 10$;
• $T_2$ follows a probability distribution with expectation $E(T_2) = 60$ and standard deviation $\sigma(T_2) = 16$;
• the random variables $T_1$ and $T_2$ are independent.

We denote $T$ the random variable giving the total waiting time before access to sports activities over the two days, expressed in minutes. Thus we have $T = T_1 + T_2$.

1. Determine the expectation $E(T)$ of the random variable $T$. Interpret the result in the context of the exercise.
2. Show that the variance $V(T)$ of the random variable $T$ is equal to 356.
3. Using the Bienaymé-Chebyshev inequality, show that, for a person randomly chosen among those who came to the multisports centre on a weekend, the probability that their total waiting time $T$ is strictly between 60 and 140 minutes is greater than 0.77.

(Probability and Statistics) Random variables $X$ and $Y$ follow normal distributions with mean 0 and variances $\sigma^2$ and $\frac{\sigma^2}{4}$ respectively, and random variable $Z$ follows the standard normal distribution. For two positive numbers $a$ and $b$, $$\mathrm{P}(|X| \leqq a) = \mathrm{P}(|Y| \leqq b)$$ Which of the following statements in are correct? [4 points] ㄱ. $a > b$ ㄴ. $\mathrm{P}\left(Z > \frac{2b}{\sigma}\right) = \mathrm{P}\left(Y > \frac{a}{2}\right)$ ㄷ. If $\mathrm{P}(Y \leqq b) = 0.7$, then $\mathrm{P}(|X| \leqq a) = 0.3$.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

A pharmaceutical company produces medicine bottles with capacity following a normal distribution with mean $m$ and standard deviation 10. When a random sample of 25 bottles is taken from the company's production, the probability that the sample mean capacity is at least 2000 is 0.9772. Using the standard normal distribution table below, what is the value of $m$? (Here, the unit of capacity is mL.) [3 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.5	0.4332
2.0	0.4772
2.5	0.4938
3.0	0.4987

(1) 2003
(2) 2004
(3) 2005
(4) 2006
(5) 2007

Let $\bar { X }$ be the sample mean obtained by randomly sampling 9 items from a population following the normal distribution $\mathrm { N } \left( 0,4 ^ { 2 } \right)$, and let $\bar { Y }$ be the sample mean obtained by randomly sampling 16 items from a population following the normal distribution $\mathrm { N } \left( 3,2 ^ { 2 } \right)$. What is the value of the constant $a$ satisfying $\mathrm { P } ( \bar { X } \geq 1 ) = \mathrm { P } ( \bar { Y } \leq a )$? [3 points]
(1) $\frac { 19 } { 8 }$
(2) $\frac { 5 } { 2 }$
(3) $\frac { 21 } { 8 }$
(4) $\frac { 11 } { 4 }$
(5) $\frac { 23 } { 8 }$

For a population following the normal distribution $\mathrm { N } \left( 20,5 ^ { 2 } \right)$, a sample of size 16 is randomly extracted and the sample mean is denoted by $\bar { X }$. What is the value of $\mathrm { E } ( \bar { X } ) + \sigma ( \bar { X } )$? [3 points]
(1) $\frac { 83 } { 4 }$
(2) $\frac { 85 } { 4 }$
(3) $\frac { 87 } { 4 }$
(4) $\frac { 89 } { 4 }$
(5) $\frac { 91 } { 4 }$

A sample of size 16 is randomly extracted from a population following a normal distribution $\mathrm { N } \left( 20,5 ^ { 2 } \right)$, and the sample mean is $\bar { X }$. What is the value of $\mathrm { E} ( \bar { X } ) + \sigma ( \bar { X } )$? [3 points]
(1) $\frac { 91 } { 4 }$
(2) $\frac { 89 } { 4 }$
(3) $\frac { 87 } { 4 }$
(4) $\frac { 85 } { 4 }$
(5) $\frac { 83 } { 4 }$

Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Suppose $p \geq 2$. Show that $$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) ^ { 1 / 2 } \leq \mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 1 / p }$$

Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Assume $1 \leq p < 2$. Show that $$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) \leq \mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 2 \theta / p } \mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { 4 } \right) ^ { ( 1 - \theta ) / 2 } .$$

Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Assume $1 \leq p < 2$. Show that there exists $\tilde { \alpha } _ { p } > 0$ such that $$\tilde { \alpha } _ { p } \mathrm { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) ^ { 1 / 2 } \leq \mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 1 / p } .$$

Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Deduce that there exists a real $\alpha _ { p }$ such that $$\alpha _ { p } \mathrm { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) ^ { 1 / 2 } \leq \mathrm { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 1 / p } .$$