LFM Stats And Pure

grandes-ecoles 2019 Q28 Expectation and Moment Inequality Proof View

In the general model of a Pólya urn ($b = c = 0$, $a = d$), using the results of questions 16 and 19, determine the expectation of $X_{n}$.

grandes-ecoles 2021 Q21 View

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 $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} \Lambda_{i,n}^{k}\right) = \frac{1}{n^{1+k/2}} \mathbb{E}\left(\operatorname{tr}\left(M_{n}^{k}\right)\right) = \frac{1}{n^{1+k/2}} \sum_{(i_{1},\ldots,i_{k}) \in \llbracket 1,n \rrbracket^{k}} \mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right).$$

grandes-ecoles 2021 Q31 Proof of Inequalities Involving Series or Sequence Terms View

Let $A > 2$. Show that, for every $(p,q) \in \mathbb{N}^{2}$, $$\mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |\Lambda_{i,n}|^{p}\right) \leqslant \frac{1}{A^{p+2q}} \mathbb{E}\left(\sum_{i=1}^{n} |\Lambda_{i,n}|^{2(p+q)}\right)$$

grandes-ecoles 2022 Q18 Probability Distribution Table Completion and Expectation Calculation View

Calculate the expectation and the variance of a variable following the distribution $\mathcal { R }$, where $\mathcal{R}$ is defined by $X ( \Omega ) = \{ - 1,1 \}$, $\mathbb { P } ( X = - 1 ) = \mathbb { P } ( X = 1 ) = \frac { 1 } { 2 }$.

grandes-ecoles 2022 Q19 Covariance Matrix and Multivariate Expectation View

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We denote by $P$ the change of basis matrix from the canonical basis of $\mathcal{M}_{n,1}(\mathbb{R})$ to an orthonormal basis formed by eigenvectors of $\Sigma_Y$. We define the discrete random variable $X = P^\top Y$, and $\Sigma_X$ is a diagonal matrix.
Deduce that the eigenvalues of $\Sigma_Y$ are all positive.

grandes-ecoles 2022 Q20 Covariance Matrix and Multivariate Expectation View

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We denote by $P$ the change of basis matrix from the canonical basis of $\mathcal{M}_{n,1}(\mathbb{R})$ to an orthonormal basis formed by eigenvectors of $\Sigma_Y$. We define the discrete random variable $X = P^\top Y = \left(\begin{array}{c} X_1 \\ \vdots \\ X_n \end{array}\right)$.
Prove that the total variance of $X$ is equal to that of $Y$.

grandes-ecoles 2022 Q21 Covariance Matrix and Multivariate Expectation View

Let $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$ be a diagonal matrix whose diagonal coefficients $\lambda_i$ are all positive. Prove the existence of a discrete random variable $Z$ with values in $\mathcal{M}_{n,1}(\mathbb{R})$ such that $\Sigma_Z = D$.

grandes-ecoles 2022 Q22 Covariance Matrix and Multivariate Expectation View

Let $A \in \mathcal{S}_n(\mathbb{R})$ be a symmetric matrix whose eigenvalues are positive. Prove the existence of a discrete random variable $Y$ with values in $\mathcal{M}_{n,1}(\mathbb{R})$ such that $\Sigma_Y = A$.

grandes-ecoles 2024 Q8c Expectation and Variance via Combinatorial Counting View

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability. We define a random variable $Z_n$ by $Z_n(\sigma) = \nu(\sigma)$.
Determine the average number of fixed points of a random permutation and its limit as $n$ tends to $+\infty$.

grandes-ecoles 2024 Q12 Convergence of Expectations or Moments View

Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Prove that $\mathbb{E}\left[X_n\right] \underset{n \rightarrow +\infty}{=} \ln(n) + \gamma + O\left(\frac{1}{n}\right)$.

grandes-ecoles 2024 Q13a Expectation and Variance via Combinatorial Counting View

Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$.
Show that $$\frac{1}{n!} \sum_{k=1}^{n} k(k-1) s(n,k) = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{1}{ij} - \sum_{i=1}^{n} \frac{1}{i^2}.$$

grandes-ecoles 2024 Q13b Expectation and Variance via Combinatorial Counting View

Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Deduce that $$\frac{1}{n!} \sum_{k=1}^{n} k^2 s(n,k) = \mathbb{E}\left[X_n\right] + \left(\sum_{i=1}^{n} \sum_{j=1}^{n} \frac{1}{ij} - \sum_{i=1}^{n} \frac{1}{i^2}\right).$$

grandes-ecoles 2024 Q14a Convergence of Expectations or Moments View

Let $n$ be a non-zero natural integer. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Show that $$\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \omega(\sigma)^2 \underset{n \rightarrow +\infty}{=} (2\gamma+1)\ln(n) + c + \ln(n)^2 + O\left(\frac{\ln(n)}{n}\right)$$ for a real number $c$ to be determined.

grandes-ecoles 2024 Q14b Convergence of Expectations or Moments View

Let $n$ be a non-zero natural integer. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Show that $$\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} (\omega(\sigma) - \ln(n))^2 \underset{n \rightarrow +\infty}{=} \ln(n) + c + O\left(\frac{\ln(n)}{n}\right).$$

grandes-ecoles 2024 Q15 View

Let $n$ be a non-zero natural integer. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Justify that there exists a real number $C > 0$ such that, for any real $\varepsilon > 0$ and any integer $n \geqslant 1$, we have $$\mathbb{P}\left(\left|X_n - \ln(n)\right| > \varepsilon \ln(n)\right) \leqslant \frac{C}{\varepsilon^2 \ln(n)}.$$

grandes-ecoles 2024 Q19 Expectation and Variance of Sums of Independent Variables View

Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Determine, for all $n \in \mathbf { N } ^ { * }$, $E \left( S _ { n } \right)$ and $V \left( S _ { n } \right)$.

grandes-ecoles 2025 QI.3 Expectation and Moment Inequality Proof View

We assume that the random variables $X_1, \ldots, X_N$ are pairwise uncorrelated, that is: $$\forall 1 \leq m, n \leq N, \quad n \neq m \Rightarrow \mathbb{E}[X_n X_m] = 0.$$ Prove that $$\mathbb{E}\left[|S_N|^2\right] \leq N.$$ Deduce that, for all $t > 0$, $$\mathbb{P}\left(|S_N| > t\sqrt{N}\right) \leq \frac{1}{t^2}$$ where $S_N := X_1 + \cdots + X_N$.

grandes-ecoles 2025 Q22 Proof of a General Conditional Expectation or Independence Property View

Let $Z$ be a random variable taking values in $\{ 0 , \ldots , M \}$. Show that:
$$\mathbf { E } [ Z ] = \sum _ { ( s , i , r ) \in E } \left( \sum _ { k = 0 } ^ { M } k \mathbf { P } \left( Z = k \mid \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right) \right) \mathbf { P } \left( \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right).$$

grandes-ecoles 2025 Q23 Existence of Expectation or Moments View

Justify that for all $n \geq 0$, the random variables $\tilde { S } _ { n } , \tilde { I } _ { n }$ and $\tilde { R } _ { n }$ as well as the random variables $\Delta \tilde { S } _ { n } , \Delta \tilde { I } _ { n }$ and $\Delta \tilde { R } _ { n }$, have finite expectation.
Here $\Delta U_n = U_{n+1} - U_n$ and the random variables take values in $\{0, \ldots, M\}$.

iran-konkur 2013 Q141 View

141- All the data in the stem-and-leaf plot below are multiplied by 3, then 40 units are subtracted from each of them. What is the new mean of the data?

Stem	\multicolumn{4}{c}{Leaf}
8	0	1	5
9	2	4	6	7
10	0	0	3	4 8

(1) 245 (2) 245 (3) 250 (4) 255
(1) $240$ (2) $245$ (3) $250$ (4) $255$

iran-konkur 2013 Q142 View

142- In 12 statistical data, the total sum of all data is 72 and the sum of their squares is 480. What is the coefficient of variation of this data?
(1) $\dfrac{1}{4}$ (2) $\dfrac{1}{9}$ (3) $\dfrac{1}{3}$ (4) $\dfrac{2}{5}$

iran-konkur 2014 Q141 View

141- Based on the following frequency distribution table, what is the value of the coefficient of variation of the data $x$?

$x - 44$	$-3$	$-1$	$1$	$3$	$5$
Frequency	$4$	$7$	$5$	$3$	$1$

(1) $0.05$ (2) $0.08$ (3) $0.1$ (4) $0.2$

iran-konkur 2014 Q142 View

142- The scores of a technical skills test for two workers $A$ and $B$ are as follows:
$A: 15, 14, 15, 16, 17, 19$
$B: 16, 14, 17, 14, 17, 18$
Which worker has greater precision?
(1) $A$ (2) $B$ (3) Equal (4) Unpredictable

iran-konkur 2015 Q141 View

141- To the statistical data shown in the frequency polygon, two data values $29$ and $32$ are added. What are the new median values?
[Figure: Frequency polygon with x-axis values 24, 27, 30, 33, 36 and y-axis (f) values approximately 8, 9, 15, 11, 12]
(1) $23$ (2) $24$
(3) $25$ (4) $26$
%% Page 26 Mathematics 120-C Page 8

iran-konkur 2015 Q142 View

142- If the mean of grouped data equals 16, to determine the fourth class frequency, which value is correct?

Class representative	12	14	16	18	20
Frequency	5	7	10	$a$	3

(1) $4/\Lambda\Delta$

(2) $4/9\Upsilon$

(3) $\Delta/\Delta\Delta$

(4) $\Delta/V4$

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LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

Solving quadratics and applications 106

Curve Sketching 295

Factor & Remainder Theorem 90

Polynomial Division & Manipulation 44

Binomial Theorem (positive integer n) 244

Function Transformations 65

Composite & Inverse Functions 249

Partial Fractions 22

Generalised Binomial Theorem 16

Generalised Binomial Theorem and Partial Fractions 3

Complex Numbers Arithmetic 283

Complex Numbers Argand & Loci 135

Permutations & Arrangements 276

Combinations & Selection 268

Measures of Location and Spread 233

Probability Definitions 413

Tree Diagrams 5

Principle of Inclusion/Exclusion 23

Independent Events 51

Conditional Probability 127

Discrete Probability Distributions 210

Binomial Distribution 107

Geometric Distribution 11

Hypergeometric Distribution 10

Negative Binomial Distribution 2

Modelling and Hypothesis Testing 38

Hypothesis test of binomial distributions 12

Data representation 29

Continuous Probability Distributions and Random Variables 30

Continuous Uniform Random Variables 15

Geometric Probability 28

Normal Distribution 96

Approximating Binomial to Normal Distribution 7

Sign Change & Interval Methods 74

Fixed Point Iteration 12