We recall that $\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^{2k+1}} = \frac{\pi^{2k+1}}{2^{2k+2}(2k)!} E_{2k}$ where $E_{2k} = v^{(2k)}(0)$ and $v(x) = \frac{1}{\cos(x)}$. We also recall that $E(g(X)) = \sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^s}$ for a random variable $X$ following the zeta distribution with parameter $s > 1$, and $g(n) = r_1(n) - r_3(n)$.
Calculate $E(g(X))$ when $X$ is a random variable following the zeta distribution with parameter 3 then with parameter 5.

We assume that the random variables $X_{ij}$ are uniformly bounded: $\exists K \in \mathbb{R} \; \forall (i,j) \in (\mathbb{N}^{\star})^{2} \; |X_{ij}| \leqslant K$.
Let $k$ be a natural integer. Justify that the random variable $\sum_{i=1}^{n} \Lambda_{i,n}^{k}$ admits an expectation and that $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} \Lambda_{i,n}^{k}\right) = \frac{1}{n^{1+k/2}} \mathbb{E}\left(\operatorname{tr}\left(M_{n}^{k}\right)\right) = \frac{1}{n^{1+k/2}} \sum_{(i_{1},\ldots,i_{k}) \in \llbracket 1,n \rrbracket^{k}} \mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right).$$

Let $k$ be a natural integer. Justify that the random variable $\sum_{i=1}^{n} \Lambda_{i,n}^{k}$ admits an expectation and that $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} \Lambda_{i,n}^{k}\right) = \frac{1}{n^{1+k/2}} \mathbb{E}\left(\operatorname{tr}\left(M_{n}^{k}\right)\right) = \frac{1}{n^{1+k/2}} \sum_{(i_{1},\ldots,i_{k}) \in \llbracket 1,n \rrbracket^{k}} \mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right).$$

We fix $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$. Show that if $V_{n}(\boldsymbol{x}) = \frac{1}{n} \sum_{i=1}^{n} |\boldsymbol{x}_{i} - \overline{\boldsymbol{x}}|^{2}$ and $V_{n}(\boldsymbol{y}) = \frac{1}{n} \sum_{i=1}^{n} |\boldsymbol{y}_{i} - \overline{\boldsymbol{y}}|^{2}$ then $$\delta(\boldsymbol{x}, \boldsymbol{y})^{2} = nV_{n}(\boldsymbol{x}) + nV_{n}(\boldsymbol{y}) - 2\sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle Z(\boldsymbol{x}, \boldsymbol{y}), R \rangle$$ where $Z(\boldsymbol{x}, \boldsymbol{y})$ is a matrix that we will specify.

Nine students in a class gave a test for 50 marks. Let $S _ { 1 } \leq S _ { 2 } \leq \cdots \leq S _ { 5 } \leq \cdots \leq S _ { 8 } \leq S _ { 9 }$ denote their ordered scores. Given that $S _ { 1 } = 20$ and $\sum _ { i = 1 } ^ { 9 } S _ { i } = 250$, let $m$ be the smallest value that $S _ { 5 }$ can take and $M$ be the largest value that $S _ { 5 }$ can take. Then the pair $( m , M )$ is given by
(A) $( 20,35 )$
(B) $( 20,34 )$
(C) $( 25,34 )$
(D) $( 25,50 )$.

Consider the given data with frequency distribution

$x _ { i }$	3	8	11	10	5	4
$f _ { i }$	5	2	3	2	4	4

Match each entry in List-I to the correct entries in List-II.
List-I
(P) The mean of the above data is
(Q) The median of the above data is
(R) The mean deviation about the mean of the above data is
(S) The mean deviation about the median of the above data is
List-II
(1) 2.5
(2) 5
(3) 6
(4) 2.7
(5) 2.4
The correct option is:
(A) $( P ) \rightarrow ( 3 )$ $( Q ) \rightarrow ( 2 )$ $( R ) \rightarrow ( 4 )$ $( S ) \rightarrow ( 5 )$
(B) $( P ) \rightarrow ( 3 )$ $( Q ) \rightarrow ( 2 )$ $( R ) \rightarrow ( 1 )$ $( S ) \rightarrow ( 5 )$
(C) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 3 )$ $( R ) \rightarrow ( 4 )$ $( S ) \rightarrow ( 1 )$
(D) $( P ) \rightarrow ( 3 )$ $( Q ) \rightarrow ( 3 )$ $( R ) \rightarrow ( 5 )$ $( S ) \rightarrow ( 5 )$

Consider the following frequency distribution:

Value	4	5	8	9	6	12	11
Frequency	5	$f _ { 1 }$	$f _ { 2 }$	2	1	1	3

Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6. For the given frequency distribution, let $\alpha$ denote the mean deviation about the mean, $\beta$ denote the mean deviation about the median, and $\sigma ^ { 2 }$ denote the variance.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I

(P) $7 f _ { 1 } + 9 f _ { 2 }$ is equal to (Q) $19 \alpha$ is equal to (R) $19 \beta$ is equal to (S) $19 \sigma ^ { 2 }$ is equal to

List-II

(1) 146
(2) 47
(3) 48
(4) 145
(5) 55

(A)	$( \mathrm { P } ) \rightarrow ( 5 )$	$( \mathrm { Q } ) \rightarrow ( 3 )$	$( \mathrm { R } ) \rightarrow ( 2 )$	$( \mathrm { S } ) \rightarrow ( 4 )$
(B)	$( \mathrm { P } ) \rightarrow ( 5 )$	$( \mathrm { Q } ) \rightarrow ( 2 )$	$( \mathrm { R } ) \rightarrow ( 3 )$	$( \mathrm { S } ) \rightarrow ( 1 )$
(C)	$( \mathrm { P } ) \rightarrow ( 5 )$	$( \mathrm { Q } ) \rightarrow ( 3 )$	$( \mathrm { R } ) \rightarrow ( 2 )$	$( \mathrm { S } ) \rightarrow ( 1 )$
(D)	$( \mathrm { P } ) \rightarrow ( 3 )$	$( \mathrm { Q } ) \rightarrow ( 2 )$	$( \mathrm { R } ) \rightarrow ( 5 )$	$( \mathrm { S } ) \rightarrow ( 4 )$

The average marks of boys in a class is 52 and that of girls is 42 . The average marks of boys and girls combined is 50 . The percentage of boys in the class is
(1) 40
(2) 20
(3) 80
(4) 60

If the mean deviation about the median of the numbers $\mathrm{a},2\mathrm{a},\ldots,50\mathrm{a}$ is 50, then $|\mathrm{a}|$ equals
(1) 3
(2) 4
(3) 5
(4) 2

If the mean of $4, 7, 2, 8, 6$ and $a$ is 7, then the mean deviation from the median of these observations is
(1) 8
(2) 5
(3) 1
(4) 3

The frequency distribution of daily working expenditure of families in a locality is as follows:

\begin{tabular}{ c } Expenditure

in ₹. $( x )$ :

& $0 - 50$ & $50 - 100$ & $100 - 150$ & $150 - 200$ & $200 - 250$ \hline

No. of

families $( f )$ :

& 24 & 33 & 37 & $b$ & 25 \hline \end{tabular}
If the mode of the distribution is Rs. 140, then the value of $b$ is
(1) 34
(2) 31
(3) 26
(4) 36

Statement 1: The variance of first $n$ odd natural numbers is $\frac { n ^ { 2 } - 1 } { 3 }$ Statement 2: The sum of first $n$ odd natural numbers is $n ^ { 2 }$ and the sum of squares of first $n$ odd natural numbers is $\frac { n \left( 4 n ^ { 2 } + 1 \right) } { 3 }$.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.
(3) Statement 1 is false, Statement 2 is true.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

Let $x_{1}, x_{2}, \ldots, x_{n}$ be $n$ observations, and let $\bar{x}$ be their arithmetic mean and $\sigma^{2}$ be their variance. Statement 1: Variance of $2x_{1}, 2x_{2}, \ldots, 2x_{n}$ is $4\sigma^{2}$. Statement 2: Arithmetic mean of $2x_{1}, 2x_{2}, \ldots, 2x_{n}$ is $4\bar{x}$.
(1) Statement 1 is false, Statement 2 is true
(2) Statement 1 is true, Statement 2 is false
(3) Statement 1 is true, Statement 2 is the correct explanation for Statement 1
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1

If the median and the range of four numbers $\{ x , y , 2 x + y , x - y \}$, where $0 < y < x < 2 y$, are 10 and 28 respectively, then the mean of the numbers is :
(1) 18
(2) 10
(3) 5
(4) 14

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given?
(1) mode
(2) variance
(3) mean
(4) median

Let $\bar { x } , M$ and $\sigma ^ { 2 }$ be respectively the mean, mode and variance of $n$ observations $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ and $d _ { i } = - x _ { i } - a , i = 1,2 , \ldots , n$, where $a$ is any number. Statement I: Variance of $d _ { 1 } , d _ { 2 } , \ldots , d _ { n }$ is $\sigma ^ { 2 }$. Statement II: Mean and mode of $d _ { 1 } , d _ { 2 } , \ldots , d _ { n }$ are $- \bar { x } - a$ and $- M - a$, respectively.
(1) Statement I and Statement II are both true
(2) Statement I and Statement II are both false
(3) Statement I is true and Statement II is false
(4) Statement I is false and Statement II is true

The variance of the first 50 even natural numbers is:
(1) 437
(2) $\frac { 437 } { 4 }$
(3) $\frac { 833 } { 4 }$
(4) 833

In a set of $2n$ distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then, the mean of the new set of observations:
(1) Increases by 2.
(2) Increase by 1.
(3) Decreases by 2.
(4) Decreases by 1.

The mean of a data set comprising of 16 observations is 16. If one of the observation value 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data is
(1) 14.0
(2) 16.8
(3) 16.0
(4) 15.8

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is:
(1) 16.8
(2) 15.8
(3) 14.0
(4) 16.0

The variance of first 50 even natural numbers is:
(1) $833$
(2) $437$
(3) $\frac{833}{4}$
(4) $833$

If the standard deviation of the numbers $2, 3, a$ and $11$ is $3.5$, then which of the following is true? (1) $3a^2 - 26a + 55 = 0$ (2) $3a^2 - 32a + 84 = 0$ (3) $3a^2 - 34a + 91 = 0$ (4) $3a^2 - 23a + 44 = 0$

The mean of 5 observations is 5 and their variance is 12.4 . If three of the observations are $1,2 \& 6$; then the value of the remaining two is :
(1) 1,11
(2) 5,5
(3) 5,11
(4) None of these

If the standard deviation of the numbers $2, 3, a$ and $11$ is $3.5$, then which of the following is true?
(1) $3a^2 - 26a + 55 = 0$
(2) $3a^2 - 32a + 84 = 0$
(3) $3a^2 - 34a + 91 = 0$
(4) $3a^2 - 23a + 44 = 0$

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is
(1) 35
(2) 40
(3) 25
(4) 30