LFM Stats And Pure

germany-abitur 2024 QA a 3 marks Finite Equally-Likely Probability Computation View

Show by calculation that the probability of winning the game is $\frac { 1 } { 4 }$.

grandes-ecoles 2015 QV.E Almost Sure Convergence and Measure-Theoretic Probability View

Let $(A_n)_{n\in\mathbb{N}}$ be a sequence of events all with probability 1.
Show that $P\left(\bigcup_{n\in\mathbb{N}}\overline{A_n}\right)=0$. What can be deduced for $P\left(\bigcap_{n\in\mathbb{N}}A_n\right)$?

grandes-ecoles 2016 Q3a Probability Bounds and Inequalities for Discrete Variables View

Show that for all $n \in \mathbb{N}$ and $\ell \geqslant 0$, $$\mathbb{P}\left(S_n \leqslant \ell\right) \leqslant \mathbb{E}\left(\exp\left(\ell - S_n\right)\right)$$ then that $$\mathbb{P}\left(S_n \leqslant \ell\right) \leqslant e^{\ell} \mathbb{E}(\exp(-X))^n$$

grandes-ecoles 2016 Q3b Probability Bounds and Inequalities for Discrete Variables View

Deduce that $\mathbb{P}\left(S_n \leqslant \ell\right)$ tends to 0 when $n \rightarrow +\infty$ and that $$\mathbb{E}(N(0,\ell)) \leqslant \frac{e^{\ell}}{1 - \mathbb{E}(\exp(-X))}$$

grandes-ecoles 2016 Q3c Probability Bounds and Inequalities for Discrete Variables View

Show that for all $x \in \mathbb{R}, \ell \geqslant 0, k \in \mathbb{N}^*$ and $n \in \mathbb{N}^*$, $$\mathbb{P}\left(S_{n-1} < x \leqslant S_n, N(x, x+\ell) \geqslant k\right) \leqslant \mathbb{P}\left(S_{n-1} < x \leqslant S_n\right) \mathbb{P}(N(0,\ell) \geqslant k)$$ then that $$\mathbb{E}(N(x, x+\ell)) \leqslant \frac{e^{\ell}}{1 - \mathbb{E}(\exp(-X))}$$

grandes-ecoles 2016 Q4a Monotonicity and Convergence of Sequences Defined via Expectations View

Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Show that for all $x \in \mathbb{R}$, the sequence $\left(f_n(x)\right)_{n \geqslant 0}$ is increasing. We denote by $f(x)$ its limit in $\mathbb{R} \cup \{+\infty\}$.

grandes-ecoles 2016 Q4b Expectation of a Function of a Discrete Random Variable View

Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Show that if $g = \mathbb{1}_{[0,K]}$, then $f(x) = \mathbb{E}(N(x-K, x))$.

grandes-ecoles 2016 Q4c Probability Bounds and Inequalities for Discrete Variables View

Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Deduce that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $$0 \leqslant f_n(x) \leqslant \|g\|_{\infty} \frac{e^K}{1 - \mathbb{E}(\exp(-X))}$$

grandes-ecoles 2016 Q4d Monotonicity and Convergence of Sequences Defined via Expectations View

Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Conclude that the sequence of functions $f_n$ converges pointwise to a positive bounded function $f$ whose support is included in $\mathbb{R}^+$.

grandes-ecoles 2016 Q5 Expectation of a Function of a Discrete Random Variable View

Let $Y$ be a discrete random variable, independent of $X$, and $\varphi : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a bounded function. Show that $$\mathbb{E}(\varphi(X, Y)) = \sum_{i=0}^{+\infty} p_i \mathbb{E}\left(\varphi\left(x_i, Y\right)\right)$$

grandes-ecoles 2016 Q6a Expectation and Variance of Sums of Independent Variables View

Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Show that for all $n \in \mathbb{N}$ and $x \in \mathbb{R}$, $$f_{n+1}(x) = g(x) + \sum_{i=0}^{+\infty} p_i f_n\left(x - x_i\right)$$

grandes-ecoles 2017 QI.A.1 Expectation and Moment Inequality Proof View

Let $U$ and $V$ be two random variables on $(\Omega, \mathcal{A}, P)$ possessing a second moment and such that $V$ is not almost surely zero. Show that $E\left(U^{2}\right) E\left(V^{2}\right) - E(UV)^{2} \geqslant 0$ and that $E\left(U^{2}\right) E\left(V^{2}\right) - E(UV)^{2} = 0$ if and only if there exists $\lambda \in \mathbb{R}$ such that $\lambda V + U$ is almost surely zero.

grandes-ecoles 2017 QII.C.1 Bounding probabilities or tail estimates via PGF View

Let $X$ and $Y$ be two random variables defined on $(\Omega, \mathcal{A}, \mathbb{P})$ and taking values in $\mathbb{N}$.
a) Show that if $A$ and $B$ are events in $\mathcal{A}$, and if $\bar{A}$ and $\bar{B}$ are their respective complementary events, then $$|\mathbb{P}(A) - \mathbb{P}(B)| \leqslant \mathbb{P}(A \cap \bar{B}) + \mathbb{P}(\bar{A} \cap B)$$
b) Deduce that, for all $t \in [-1,1], \left|G_{X}(t) - G_{Y}(t)\right| \leqslant 2\mathbb{P}(X \neq Y)$.

grandes-ecoles 2018 QII.2 Probability Inequality and Tail Bound Proof View

Let $k$ be a strictly positive integer and $U_{1}, \ldots, U_{k}$ a sequence of $k$ random variables taking values in $\{-1,1\}$, independent and uniformly distributed. We also denote $$S_{k} = \sum_{i=1}^{k} U_{i}$$ Let $\varphi(\lambda) = \ln\left(\mathbb{E}\left[e^{\lambda U_{1}}\right]\right)$.
Let $t \in \mathbb{R}$. Show that for all $\lambda > 0$, we have the inequality $$\mathbb{P}\left(S_{k} \geqslant t\right) \leqslant \exp(k\varphi(\lambda) - \lambda t).$$

grandes-ecoles 2018 Q6 Expectation and Moment Inequality Proof View

Let $p$ and $q$ be two strictly positive reals such that $\frac{1}{p} + \frac{1}{q} = 1$. Deduce that if $X$ and $Y$ are two real-valued random variables on the finite probability space $(\Omega, \mathcal{A}, \mathbb{P})$, then
$$\mathbb{E}(|XY|) \leqslant \mathbb{E}(|X|^{p})^{1/p} \mathbb{E}(|Y|^{q})^{1/q}$$
You may first prove this result when $\mathbb{E}(|X|^{p}) = \mathbb{E}(|Y|^{q}) = 1$.

grandes-ecoles 2018 Q7 Expectation of a Function of a Discrete Random Variable View

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. Let $(A_{1}, \ldots, A_{m})$ be a complete system of events with non-zero probabilities. Show that
$$\mathbb{E}(X) = \sum_{i=1}^{m} \mathbb{P}(A_{i}) \cdot \mathbb{E}(X \mid A_{i})$$

grandes-ecoles 2018 Q10 Probability Inequality and Tail Bound Proof View

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Let $\delta$ be a real such that $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Justify that, for all reals $t$,
$$\mathbb{P}(|X + \delta| \geqslant t) \leqslant \mathbb{P}(|X| \geqslant t - |\delta|)$$

grandes-ecoles 2018 Q11 Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) View

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Let $\delta$ be a real such that $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Show that, for all reals $t$,
$$-b(t - |\delta|)^{2} \leqslant a - \frac{1}{2}bt^{2}$$

grandes-ecoles 2018 Q12 Probability Inequality and Tail Bound Proof View

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Let $\delta$ be a real such that $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Deduce that for all reals $t$ such that $t \geqslant |\delta|$, we have
$$\mathbb{P}(|X + \delta| \geqslant t) \leqslant a \exp(a) \exp\left(-\frac{1}{2}bt^{2}\right)$$

grandes-ecoles 2018 Q13 Probability Inequality and Tail Bound Proof View

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Let $\delta$ be a real such that $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Justify that the inequality
$$\mathbb{P}(|X + \delta| \geqslant t) \leqslant a \exp(a) \exp\left(-\frac{1}{2}bt^{2}\right)$$
remains valid if $0 \leqslant t < |\delta|$.

grandes-ecoles 2018 Q14 Probability Inequality and Tail Bound Proof View

Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. The objective of this part is to show, for any non-empty closed convex set $C$ of $E$,
$$\mathbb{P}(X \in C) \cdot \mathbb{E}\left(\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right) \leqslant 1 \tag{II.1}$$
Handle the case where $C$ is a closed convex set of $E$ that does not meet $X(\Omega)$.

grandes-ecoles 2018 Q16 Expectation and Moment Inequality Proof View

Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. We assume that $C$ is a closed convex set of $E$ that meets $X(\Omega)$ in a single vector $u$. Deduce the expectation of $\exp\left(\frac{1}{8} d(X, u)^{2}\right)$ and show that it is less than or equal to $2^{n}$.

grandes-ecoles 2018 Q17 Probability Inequality and Tail Bound Proof View

Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. We assume that $C$ is a closed convex set of $E$ that meets $X(\Omega)$ in a single vector $u$. Justify that $d(X, C) \leqslant d(X, u)$ and deduce inequality
$$\mathbb{P}(X \in C) \cdot \mathbb{E}\left(\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right) \leqslant 1 \tag{II.1}$$
in this case.

grandes-ecoles 2018 Q18 Probability Inequality and Tail Bound Proof View

Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. We assume that $C$ is a closed convex set of $E$ such that $C \cap X(\Omega)$ contains at least two elements. We propose to prove inequality
$$\mathbb{P}(X \in C) \cdot \mathbb{E}\left(\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right) \leqslant 1 \tag{II.1}$$
by induction on the dimension $n$ of $E$. Handle the case $n = 1$.

grandes-ecoles 2018 Q19 Verification of Probability Measure or Inner Product Properties View

Let $n$ be an integer such that $n \geqslant 2$. We denote by $E' = \operatorname{Vect}(e_{1}, \ldots, e_{n-1})$ and by $\pi$ the orthogonal projection onto $E'$
$$\pi : \left\lvert \, \begin{aligned} E & \rightarrow E' \\ \sum_{i=1}^{n} x_{i} e_{i} & \mapsto \sum_{i=1}^{n-1} x_{i} e_{i} \end{aligned} \right.$$
We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $H_{t}$ the affine hyperplane $E' + te_{n}$ and $C_{t} = \pi(C \cap H_{t})$.
Show, for $x' \in E'$ and $t \in \{-1, 1\}$, that $x' \in C_{t} \Longleftrightarrow x' + te_{n} \in C$.

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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