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 $\mathrm{E}(X)$.

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 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) $\frac { 15 } { 2 }$