Exercise 0.29 (Skorokhod representation of scalar variables)

Question

Let $U$ be a uniform random variable taking values in $[0,1]$, and let $F:\mathbb{R}	o[0,1]$ be another cumulative distribution function. Show that the random variables
$$X^-:=\sup\left\{y\in\mathbb{R}:F(y) < Uight\}$$
and
$$X^+:=\inf\left\{y\in\mathbb{R}:F(y) \geq Uight\}$$
are indeed random variables.

Elu Thingol · Accepted Answer

Let $\Omega$ be the underlying randomness space. In this case, the variables $X^-$ and $X^+$ collapse to $U$. To see why, pick an arbitrary $\omega \in \Omega$:
\begin{align*}
X^-(\Omega) &= \sup \left \{y \in [0,1]: F(y) < U(\omega)ight\}\
&= \sup\left\{y\in [0,1]: P(U\leq y) < U(\omega)ight\}\
&= \sup\left\{y\in [0,1]: y < U(\omega)ight\}\
&= U(\omega).
\end{align*}

Exercise 0.29 (Skorokhod representation of scalar variables)

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Exercise 0.29 (Skorokhod representation of scalar variables)

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