I am being asked to solve the following problem:
Assume you have a sequence of i.i.d. (independent identically distributed) random variables, $X_1, X_2, \dots,$ on a probability space $(\Omega,\mathcal{F},P)$ with $P(X_n=1)=P(X_n=-1)=1/2$. Given a distribution function, $F$, use the $X_n$'s to construct a sequence of i.i.d. random variables, $Y_1, Y_2, \dots,$ with distribution function $F$. [Hint: First show that $U=\sum_{n=1}^{\infty} 2^{-n}X_n$ is a Uniform$([0,1])$ random variable, then use $U$ to find one random variable with distribution function F.]
If it helps this was the part (b) of the problem. Prior to this, in part (a) we were asked to solve:
Let $U$ be a Uniform($[0,1]$) random variable (i.e., the distribution of $U$ is the Lebesgue measure on $[0,1]$). Define $X_n=\lfloor 2^n U \rfloor, \ n= 1,2,\dots,$ to be the $n^{\text{th}}$ digit in the binary expansion of $U$
($\lfloor x \rfloor$ is the greatest integer less than or equal to $x$). Show that $X_1, X_2, \dots $ are i.i.d. random variables. Note i.i.d. stands for independent identically distributed.
So in any help you provide feel free to use this result, without proof.
I am really lost, I don't know how to proceed.
I would appreciate any help.
Thanks!