Let $\epsilon_1, \epsilon_2, \dots \in \{0,1\}$ be random variables that are independent and identically distributed copies of the Bernoulli random variable with expectation $1/2$. \begin{enumerate} \item Show that the random variable $\sum_{n=1}^\infty 2^{-n} \epsilon_n$ is uniformly distributed on the unit interval $[0,1]$. \item Show that the random variable $\sum_{n=1}^\infty 2 \times 3^{-n} \epsilon_n$ has the distribution of Cantor measure. \end{enumerate}

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Exercise 2.27 (Bernoulli random variables)

Let $𝜖_{1}, 𝜖_{2}, \dots \in {0, 1}$ be random variables that are independent and identically distributed copies of the Bernoulli random variable with expectation $1 ∕ 2$ .

1.: Show that the random variable $\sum_{n = 1}^{\infty} 2^{- n} 𝜖_{n}$ is uniformly distributed on the unit interval $[0, 1]$ .
2.: Show that the random variable $\sum_{n = 1}^{\infty} 2 \times 3^{- n} 𝜖_{n}$ has the distribution of Cantor measure.

Exercise 2.27 (Bernoulli random variables)

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