grandes-ecoles 2016 QII.B.1

grandes-ecoles · France · centrale-maths2__mp Discrete Random Variables Expectation of a Function of a Discrete Random Variable

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$ and $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$.
Show that $E\left(S_{n}^{4}\right) = 3n^{2} - 2n$ for all $n \in \mathbb{N}^{*}$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$,
$$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$
For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$ and $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$.

Show that $E\left(S_{n}^{4}\right) = 3n^{2} - 2n$ for all $n \in \mathbb{N}^{*}$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2