Example of independent and identically distributed random variables

independencemeasure-theoryprobability theoryrandom variables

Consider the probability space $(\Bbb [0, 1], \mathcal{B}, \mu),$ where $\mathcal{B}$ denotes the Borel sigma algebra in $[0, 1],$ and $\mu$ is the standard Lebesgue measure restricted to $[0, 1]$. Could you please provide an example (mathematically defined) of two absolutely continuous random variables $X_1, X_2 : ([0, 1], \mathcal{B}, \mu) \rightarrow \Bbb R$ that are independent and identically distributed?

Best Answer

Let, for all $x \in [0,1]$, $X_1(x) := \sum^{\infty}_{n=1} \lfloor \frac{x}{2^{2n}} \rfloor \frac{1}{2^n}$ and $X_2(x) := \sum^{\infty}_{n=1} \lfloor \frac{x}{2^{2n-1}}\rfloor \frac{1}{2^n}$.

I leave you, as an exercise, to prove that $X_1$ and $X_2$ are indeed iid.

Hints:

  1. For every $k \in \mathbb{N}$, and $n \in \mathbb{N}^*$, compute $X^{-1}_1([\frac{k}{2^n},\frac{k+1}{2^n}))$ and $X^{-1}_2([\frac{k}{2^n},\frac{k+1}{2^n}))$. Each of these should be a dyadic interval.

  2. Deduce from this that $X_1$ and $X_2$ are both uniform and independent.

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