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

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad F_n(x) = \mathbb{P}(Y_n \leqslant x).$$ We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Show $$\forall n \in \mathbb{N}^{\star},\, \forall x \in D_n, \quad F_n(x) = x + \frac{1}{2^n}.$$

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

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad G_n(x) = \mathbb{P}(Y_n < x).$$ We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Show $$\forall n \in \mathbb{N}^{\star},\, \forall x \in D_n, \quad G_n(x) = x.$$