To promote the organic products of his store, a store manager decides to organize a game that consists, for a customer, of filling a basket with a certain mass of apricots from organic farming. It is announced that the customer wins the contents of the basket if the mass of apricots deposited is between 3.2 and 3.5 kilograms.
The mass of fruit in kg, placed in the basket by customers, can be modeled by a random variable $X$ following the probability distribution with density $f$ defined on the interval $[3; 4]$ by: $$f(x) = \frac{2}{(x-2)^2}$$
Reminder: a probability density function on the interval $[a; b]$ is any function $f$ defined, continuous and positive on $[a; b]$, such that the integral of $f$ over $[a; b]$ is equal to 1.

1. Verify that the function $f$ previously defined is indeed a probability density function on the interval $[3;4]$.
2. The store announces: ``One customer in three wins the basket!''. Is this announcement accurate?
3. The purpose of this question is to calculate the mathematical expectation $\mathrm{E}(X)$.

Let $f ( x ) = \left| \frac { \sin x } { x } \right| ^ { 1.001 }$ for $x \neq 0$ and $f ( 0 ) = L$ such that $f$ is continuous. Let $I ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
Statements
(21) $L = 1.001$ (22) $I ( 0.001 ) > 0.001$. (23) As $x \rightarrow \infty$ the limit of $I ( x )$ is greater than 1001 (possibly $\infty$). (24) The function $I ( x )$ is NOT differentiable at infinitely many points.

Let $f : \mathbb { R } \longrightarrow ( 0 , \infty )$ be an infinitely differentiable function with $\int _ { - \infty } ^ { \infty } f ( t ) d t = 1$. Pick the correct statement(s) from below.
(A) $f ( t )$ is bounded.
(B) $\lim _ { | t | \rightarrow \infty } f ^ { \prime } ( t ) = 0$.
(C) There exists $t _ { 0 } \in \mathbb { R }$ such that $f \left( t _ { 0 } \right) \geq f ( t )$ for all $t \in \mathbb { R }$.
(D) $f ^ { \prime \prime } ( a ) = 0$ for some $a \in \mathbb { R }$.

Pick the correct statement(s) from below.
(A) If $f$ is continuous and bounded on $( 0,1 )$, then $f$ is uniformly continuous on $( 0,1 )$.
(B) If $f$ is uniformly continuous on $( 0,1 )$, then $f$ is bounded on $( 0,1 )$.
(C) If $f$ is continuous on $( 0,1 )$ and $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ and $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ exists, then $f$ is uniformly continuous on $( 0,1 )$.
(D) Product of a continuous and a uniformly continuous function on $[ 0,1 ]$ is uniformly continuous.

Let $X$ be the metric space of real-valued continuous functions on the interval $[ 0,1 ]$ with the ``supremum distance'': $$d ( f , g ) = \sup \{ | f ( x ) - g ( x ) | : x \in [ 0,1 ] \} \text { for all } f , g \in X$$ Let $Y = \{ f \in X : f ( [ 0,1 ] ) \subset [ 0,1 ] \}$ and $Z = \left\{ f \in X : f ( [ 0,1 ] ) \subset \left[ 0 , \frac { 1 } { 2 } \right) \cup \left( \frac { 1 } { 2 } , 1 \right] \right\}$. Pick the correct statement(s) from below.
(A) $Y$ is compact.
(B) $X$ and $Y$ are connected.
(C) $Z$ is not compact.
(D) $Z$ is path-connected.

Let $X : = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid z \leq 0 \right.$, or $\left. x , y \in \mathbb { Q } \right\}$ with subspace topology. Pick the correct statement(s) from below.
(A) $X$ is not locally connected but path connected.
(B) There exists a surjective continuous function $X \longrightarrow \mathbb { Q } _ { \geq 0 }$ (the set of non-negative rational numbers, with the subspace topology of $\mathbb { R }$ ).
(C) Let $S$ be the set of all points $p \in X$ having a compact neighbourhood (i.e. there exists a compact $K \subset X$ containing $p$ in its interior). Then $S$ is open.
(D) The closed and bounded subsets of $X$ are compact.

Let $Q$ be the space of infinite sequences $$\mathbf { x } : = \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots \right)$$ of real numbers $x _ { n } \in [ 0,1 ]$, with the product topology coming from the identification $Q = [ 0,1 ] ^ { \mathbb { N } }$. ($[ 0,1 ]$ has the euclidean topology.) Let $S : Q \longrightarrow \mathbb { R }$ be the map $$S ( \mathbf { x } ) : = \sum _ { n } \frac { x _ { n } } { n ^ { 2 } } .$$ (A) Let $Q _ { 2 } : = \left\{ \left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \left\lvert \, 0 \leq y _ { n } \leq \frac { 1 } { n } \right. \right\}$. Show that $Q _ { 2 }$ is compact.
(B) Let $D : Q _ { 2 } \longrightarrow Q$ be the map $$\left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \mapsto \left( y _ { 1 } , 2 y _ { 2 } , \ldots , n y _ { n } , \ldots \right)$$ Show that $D$ is a homeomorphism. (Hint: first show that $Q$ is Hausdorff.)
(C) Show that $S \circ D : Q _ { 2 } \longrightarrow \mathbb { R }$ is continuous. (Hint: Show that there is a suitable inner-product space $( L , \langle - , - \rangle )$ and a vector $\mathbf { a } \in L$ such that $( S \circ D ) ( \mathbf { x } ) = \langle \mathbf { x } , \mathbf { a } \rangle$ for each $\mathbf { x } \in Q _ { 2 }$.)
(D) Show that $S$ is continuous.

The following is a probability distribution table of a certain population.

$X$	1	2	3	Total
$\mathrm { P } ( X )$	0.5	0.3	0.2	1

When a sample of size 2 is drawn with replacement from this population, the probability distribution table of the sample mean $\bar { X }$ is as follows.

$\bar { X }$	1	1.5	2	2.5	3
Frequency	1	$a$	$b$	2	1
$\mathrm { P } ( \bar { X } )$	0.25	$c$	$d$	0.12	0.04

Find the value of $100 ( b + c )$. [4 points]

A continuous random variable $X$ has a range of $0 \leqq X \leqq 3$, and the graph of its probability density function is as follows. When $\mathrm { P } ( m \leqq X \leqq 2 ) = \mathrm { P } ( 2 \leqq X \leqq 3 )$, what is the value of $m$? (Here, $0 < m < 2$.) [3 points]
(1) $\frac { \sqrt { 2 } } { 2 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) 1
(4) $\sqrt { 2 }$
(5) $\sqrt { 3 }$

For two positive numbers $a , b$, a continuous random variable $X$ has a range of $0 \leqq X \leqq a$, and the graph of the probability density function is as shown. When $\mathrm { P } \left( 0 \leqq X \leqq \frac { a } { 2 } \right) = \frac { b } { 2 }$, find the value of $a ^ { 2 } + 4 b ^ { 2 }$. [4 points]

A continuous random variable $X$ has a range of $0 \leqq X \leqq 3$, and the probabilities $\mathrm { P } ( X \leqq 1 )$ and $\mathrm { P } ( X \leqq 2 )$ are the two roots of the quadratic equation $6 x ^ { 2 } - 5 x + 1 = 0$. What is the value of the probability $\mathrm { P } ( 1 < X \leqq 2 )$? [3 points]
(1) $\frac { 1 } { 12 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 5 } { 12 }$

A continuous random variable $X$ has a range of $0 \leqq X \leqq 4$, and the graph of the probability density function of $X$ is as shown in the figure. Find the value of $100 \mathrm { P } ( 0 \leqq X \leqq 2 )$. [4 points]

A continuous random variable $X$ defined on the interval $[ 0,1 ]$ has probability density function $f ( x )$. The mean of $X$ is $\frac { 1 } { 4 }$ and $\int _ { 0 } ^ { 1 } ( a x + 5 ) f ( x ) d x = 10$. Find the value of the constant $a$. [4 points]

A continuous probability density function is defined on the closed interval $[ 0 , a ]$ for a random variable $X$. When the random variable $X$ satisfies the following conditions, what is the value of the constant $k$? [4 points] (가) For all $x$ where $0 \leq x \leq a$, $\mathrm { P } ( 0 \leq X \leq x ) = k x ^ { 2 }$. (나) $\mathrm { E } ( X ) = 1$
(1) $\frac { 9 } { 16 }$
(2) $\frac { 4 } { 9 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 9 }$
(5) $\frac { 1 } { 16 }$

For a continuous random variable $X$ that takes all real values in the interval $[ 0,3 ]$, the graph of the probability density function of $X$ is shown in the figure. If $\mathrm { P } ( 0 \leq X \leq 2 ) = \frac { q } { p }$, find the value of $p + q$. (Here, $k$ is a constant, and $p$ and $q$ are coprime natural numbers.) [4 points]

For a continuous random variable $X$ that takes all real values in the closed interval $[ 0,1 ]$, the probability density function is $$f ( x ) = k x \left( 1 - x ^ { 3 } \right) \quad ( 0 \leq x \leq 1 )$$ Find the value of $24 k$. (Here, $k$ is a constant.) [3 points]

A continuous random variable $X$ has range $0 \leq X \leq 2$, and the graph of the probability density function of $X$ is shown in the figure. What is the value of $\mathrm { P } \left( \frac { 1 } { 3 } \leq X \leq a \right)$? (Here, $a$ is a constant.) [3 points] [Figure]
(1) $\frac { 11 } { 16 }$
(2) $\frac { 5 } { 8 }$
(3) $\frac { 9 } { 16 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 7 } { 16 }$

Two continuous random variables $X$ and $Y$ have ranges $0 \leq X \leq 6$ and $0 \leq Y \leq 6$, with probability density functions $f ( x )$ and $g ( x )$ respectively. The graph of the probability density function $f ( x )$ of random variable $X$ is shown in the figure.
For all $x$ with $0 \leq x \leq 6$, $$f ( x ) + g ( x ) = k \text{ (where } k \text{ is a constant)}$$
When $\mathrm { P } ( 6 k \leq Y \leq 15 k ) = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

A continuous random variable $X$ has a range of $0 \leq X \leq a$, and the graph of the probability density function of $X$ is as shown in the figure. When $\mathrm { P } ( X \leq b ) - \mathrm { P } ( X \geq b ) = \frac { 1 } { 4 }$ and $\mathrm { P } ( X \leq \sqrt { 5 } ) = \frac { 1 } { 2 }$, what is the value of $a + b + c$? (Here, $a$, $b$, and $c$ are constants.) [4 points]
(1) $\frac { 11 } { 2 }$
(2) 6
(3) $\frac { 13 } { 2 }$
(4) 7
(5) $\frac { 15 } { 2 }$

Throughout this part, $\lambda$ is a real number belonging to the interval $]0,1[$ and $f, g, h$ are functions in $C^{0}(\mathbb{R}, \mathbb{R}_{+})$ that are integrable and satisfy the following inequality $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \quad h(\lambda x + (1-\lambda) y) \geq f(x)^{\lambda} g(y)^{1-\lambda}.$$ In questions 3), 4) and 5) we additionally assume that $f$ and $g$ are strictly positive, that is, for all real $x$, $f(x) > 0$ and $g(x) > 0$.
We denote $F = \int_{-\infty}^{+\infty} f(x)\,dx$ and $G = \int_{-\infty}^{+\infty} g(x)\,dx$. Show that for all $t$ in the interval $]0,1[$ there exists a unique real number denoted $u(t)$ and a unique real number denoted $v(t)$ such that $$\frac{1}{F} \int_{-\infty}^{u(t)} f(x)\,dx = t, \quad \frac{1}{G} \int_{-\infty}^{v(t)} g(x)\,dx = t$$ (One may study the variations of the function: $u \mapsto \frac{1}{F} \int_{-\infty}^{u} f(x)\,dx$).

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Establish that $f$ is in $\mathcal{B}_1$.

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Show that $\hat{f}$ is defined on $\mathbb{R}^2$ with $\hat{f}(q,\theta) = \frac{\pi}{\sqrt{1+q^2}}$.

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
We set $R(q) = \frac{1}{2\pi} \int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta$. Prove that $q \mapsto \frac{R^{\prime}(q)}{q}$ is integrable on $]0, +\infty[$ and that $$f(0,0) = -\frac{1}{\pi} \int_0^{+\infty} \frac{R^{\prime}(q)}{q}\,\mathrm{d}q$$ One may, to compute this last integral, use the change of variable $q = \operatorname{sh}(u)$.

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Is the function $\frac{\partial f}{\partial x}$ in $\mathcal{B}_2$?

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
For $r \in \mathbb{R}^+$, compute $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.