grandes-ecoles 2019 Q19

grandes-ecoles · France · centrale-maths2__mp Continuous Probability Distributions and Random Variables Distribution of Transformed or Combined Random Variables

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}.$$
Justify $$\forall n \in \mathbb{N}^{\star}, \quad \mathbb{P}(Y_n \in [0,1[) = 1.$$

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set
$$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}.$$

Justify
$$\forall n \in \mathbb{N}^{\star}, \quad \mathbb{P}(Y_n \in [0,1[) = 1.$$

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