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Let $U=(U_{1},U_{2},…)$ represent an infinite sequence of coin tosses, with $U_{i}=1$ if the $i$ th toss is heads and $U_{i}=0$ if it is tails. Suppose the coin tosses are independent, and that the probability of heads is $p∈(0,1)$ and the probability of tails is $q=1−p$. Given such a sequence we can define a real-valued random variable $X=f(U)$, taking values in the interval $[0,1]$, by $f(U)=∑_{i=1}2_{i}U_{i} .$ We may think of $U$ as a binary expansion of $X$ (though in fact some $x∈[0,1]$ do not have a unique binary expansion, to wit, $0.1=0.01111…$.. Define the cumulative distribution function $F(x)=P(X⩽x).$ For most values of $p$ the function $F$ is pathological, but it does have some interesting properties.
Question 2 Suppose that $x=∑_{i=1}2_{i}x_{i} $ for some $n∈N$ and some sequence $x_{1},…,x_{n}$. (When this is so, we say $x$ has a finite binary expansion.) Find a formula for $F(x)$.

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