What is the domain of definition $\mathcal{D}$ of the function $\Gamma$, where for $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$?
For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. For all $x \in \mathcal{D}$, express $\Gamma(x+1)$ in terms of $x$ and $\Gamma(x)$. Deduce from this, for all $x \in \mathcal{D}$ and all $n \in \mathbb{N}^{*}$, an expression for $\Gamma(x+n)$ in terms of $x$, $n$ and $\Gamma(x)$, as well as the value of $\Gamma(n)$ for all $n \geqslant 1$.
For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show the existence of the two integrals $\int_{0}^{+\infty} e^{-t^{2}} \mathrm{~d}t$ and $\int_{0}^{+\infty} e^{-t^{4}} \mathrm{~d}t$ and express them using $\Gamma$.
Let $a$ and $b$ be two real numbers such that $0 < a < b$. Show that, for all $t > 0$ and all $x \in [a, b]$, $$t^{x} \leqslant \max\left(t^{a}, t^{b}\right) \leqslant t^{a} + t^{b}$$
For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma$ is of class $\mathcal{C}^{\infty}$ on $\mathcal{D}$. Let $k \in \mathbb{N}^{*}$ and $x \in \mathcal{D}$. Express $\Gamma^{(k)}(x)$, the $k$-th derivative of $\Gamma$ at point $x$, in the form of an integral.
For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma^{\prime}$ vanishes at a unique real number $\xi$ whose integer part will be determined.
For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Deduce the variations of $\Gamma$ on $\mathcal{D}$. Specify in particular the limits of $\Gamma$ at 0 and at $+\infty$. Also specify the limits of $\Gamma^{\prime}$ at 0 and at $+\infty$. Sketch the graph of $\Gamma$.
For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, where $\mathrm{i}$ denotes the complex number with modulus 1 and argument $\pi/2$. Show that the function $F : \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto F(x) \end{aligned}$ is defined and of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$. Let $k$ be a non-zero natural number and let $x$ be a real number. Give an integral expression for $F^{(k)}(x)$, the $k$-th derivative of $F$ at $x$. Specify $F(0)$.
For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Show that in a neighbourhood of $x = 0$, the function $F$ can be written in the form $$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$ where $c_{n}$ is the value of Gamma at a point to be specified. Express $c_{n}$ in terms of $n$ and $c_{0}$. What is the radius of convergence of the power series appearing on the right-hand side of $(S)$?
For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, and $$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$ We admit that $\Gamma(x) \underset{x \rightarrow +\infty}{\sim} \sqrt{2\pi} x^{(x-1/2)} \mathrm{e}^{-x}$. Investigate whether the series on the right-hand side of $(S)$ converges absolutely when $|x| = R$, where $R$ is the radius of convergence.
For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Let $R(x)$ be the real part and $I(x)$ be the imaginary part of $F(x)$. Determine, in a neighbourhood of 0, the Taylor expansion of $R(x)$ to order 3 and of $I(x)$ to order 4.
For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Prove that $F$ satisfies on $\mathbb{R}$ a differential equation of the form $F^{\prime} + AF = 0$, where $A$ is a function to be specified.
For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, and $F$ satisfies $F^{\prime} + AF = 0$ on $\mathbb{R}$. Deduce an expression for $F(x)$. You may start by differentiating the function $x \mapsto -\frac{1}{8} \ln\left(1 + x^{2}\right) + \frac{\mathrm{i}}{4} \arctan x$.
Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ We denote $G_{X}(t) = \mathrm{E}\left(t^{X}\right) = \sum_{k=0}^{\infty} \mathrm{P}(X = k) t^{k}$ (generating series of the random variable $X$). Let $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Determine $G_{X}(t)$.
Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ Let $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Calculate the expectation $\mathrm{E}(X)$, the variance $V(X)$ and the standard deviation of $X$.
Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ Let $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Let $\mu$ be a strictly positive real number. Let $Y$ be a random variable following the Poisson distribution $\mathcal{P}(\mu)$ and such that $X$ and $Y$ are independent. Determine the distribution of $X + Y$.
Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables, with distribution $\mathcal{P}(\lambda)$. For all integers $n \geqslant 1$, determine the distribution of $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$.
Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables, with distribution $\mathcal{P}(\lambda)$, and $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$. Determine the expectation and standard deviation of the random variables $S_{n}$ and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$.
Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables, with distribution $\mathcal{P}(\lambda)$, $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$, and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$. Show that, for all $\varepsilon > 0$, there exists a real number $c(\varepsilon)$ such that, if $c \geqslant c(\varepsilon)$ and $n \in \mathbb{N}^{*}$, we have $\mathrm{P}\left(\left|T_{n}\right| \geqslant c\right) \leqslant \varepsilon$.
We consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$ for all $x \in \mathbb{R}$. Show that there exists a real number $M > 0$ such that $f$ is an $M$-Lipschitz function.
We fix two real numbers $a$ and $b$ such that $a < b$. For all integers $n \geqslant 1$ such that $a + \sqrt{n\lambda} > 0$, we define $$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$ For $k \in \mathbb{Z}$, we define $x_{k,n} = \frac{k - n\lambda}{\sqrt{n\lambda}}$. We consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$ for all $x \in \mathbb{R}$, which is $M$-Lipschitz for some $M > 0$. a) Show that, if $x, h \in \mathbb{R}$ and $h > 0$, then $\left| hf(x) - \int_{x}^{x+h} f(t) \mathrm{d}t \right| \leqslant M \frac{h^{2}}{2}$. b) Deduce from this, when $I_{n}$ is non-empty, an upper bound for $$\left| \frac{1}{\sqrt{n\lambda}} \sum_{k \in I_{n}} f\left(x_{k,n}\right) - \int_{x_{p,n}}^{x_{q+1,n}} f(t) \mathrm{d}t \right|$$ where $p$ is the smallest element of $I_{n}$ and $q$ is the largest. c) Show that $$\lim_{n \rightarrow +\infty} \frac{1}{\sqrt{n\lambda}} \sum_{k \in I_{n}} f\left(x_{k,n}\right) = \int_{a}^{b} f(x) \mathrm{d}x$$
We fix two real numbers $a$ and $b$ such that $a < b$. For all integers $n \geqslant 1$ such that $a + \sqrt{n\lambda} > 0$, we define $$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$ For $k \in \mathbb{Z}$, we define $x_{k,n} = \frac{k - n\lambda}{\sqrt{n\lambda}}$. We consider the function $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$. For all $k \in I_{n}$, we denote $y_{k,n} = \left(1 - \frac{x_{k,n}}{k}\sqrt{n\lambda}\right)^{k} \exp\left(x_{k,n}\sqrt{n\lambda}\right)$. Let $\varepsilon > 0$. Prove the existence of an integer $N(\varepsilon)$ such that, for all $n \geqslant N(\varepsilon)$ and all $k \in I_{n}$, the following inequalities are satisfied: a) $\frac{1-\varepsilon}{\sqrt{2\pi}} \frac{1}{\sqrt{n\lambda}} y_{k,n} \leqslant \mathrm{e}^{-n\lambda} \frac{(n\lambda)^{k}}{k!} \leqslant \frac{1+\varepsilon}{\sqrt{2\pi}} \frac{1}{\sqrt{n\lambda}} y_{k,n}$; We will use Stirling's formula $n! \underset{n \rightarrow +\infty}{\sim} \sqrt{2\pi n} \left(\frac{n}{\mathrm{e}}\right)^{n}$. b) $(1-\varepsilon) f\left(x_{k,n}\right) \leqslant y_{k,n} \leqslant (1+\varepsilon) f\left(x_{k,n}\right)$.
We fix two real numbers $a$ and $b$ such that $a < b$. For all integers $n \geqslant 1$ such that $a + \sqrt{n\lambda} > 0$, we define $$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$ Express, in the form of an integral, $\lim_{n \rightarrow +\infty} \sum_{k \in I_{n}} \frac{(n\lambda)^{k}}{k!} \mathrm{e}^{-n\lambda}$.
We fix two real numbers $a$ and $b$ such that $a < b$. For all integers $n \geqslant 1$ such that $a + \sqrt{n\lambda} > 0$, we define $$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$ Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables with distribution $\mathcal{P}(\lambda)$, $S_{n} = X_{1} + \cdots + X_{n}$, and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$. Compare $\mathrm{P}\left(a \leqslant T_{n} \leqslant b\right)$ and $\sum_{k \in I_{n}} \mathrm{P}\left(S_{n} = k\right)$.
Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables with distribution $\mathcal{P}(\lambda)$, $S_{n} = X_{1} + \cdots + X_{n}$, and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$. Determine the limits, when $n \rightarrow +\infty$, of $$\mathrm{P}\left(T_{n} \geqslant a\right), \quad \mathrm{P}\left(T_{n} = a\right), \quad \mathrm{P}\left(T_{n} > a\right) \quad \text{and} \quad \mathrm{P}\left(T_{n} \leqslant b\right)$$
Recall that a random variable $X$ follows the Poisson distribution $\mathcal{P}(\lambda)$ if $\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$ for all $n \in \mathbb{N}$. Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables with distribution $\mathcal{P}(\lambda)$, $S_{n} = X_{1} + \cdots + X_{n}$, and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$. Determine an equivalent, when $n \rightarrow +\infty$, of $$A_{n} = \sum_{k=0}^{\lfloor n\lambda \rfloor} \frac{(n\lambda)^{k}}{k!} \quad \text{and} \quad B_{n} = \sum_{k=\lfloor n\lambda \rfloor + 1}^{+\infty} \frac{(n\lambda)^{k}}{k!}$$ where $\lfloor t \rfloor$ denotes the integer part of the real number $t$. We will interpret $\mathrm{e}^{-n\lambda} A_{n}$ as the probability of an event related to $S_{n}$ and thus to $T_{n}$.
We assume $\lambda < 1$, and $D_{n} = \sum_{k=n+1}^{+\infty} \frac{(n\lambda)^{k}}{k!}$. Using Taylor's formula with integral remainder, deduce an equivalent of $D_{n}$ when $n \rightarrow +\infty$.
If $\lambda > 1$, and $C_{n} = \sum_{k=0}^{n} \frac{(n\lambda)^{k}}{k!}$, determine an equivalent of $C_{n}$ when $n \rightarrow +\infty$. Consider the integral $\frac{1}{n!} \int_{-\infty}^{0} (r - t)^{n} \mathrm{e}^{t} \mathrm{~d}t$ and choose the real number $r$ appropriately.