# Let (F: mathbb{R} ightarrow[0,1]) be a distribution function. a) Show that there exists a probability space ((Omega,

## Question:

Let \(F: \mathbb{R} ightarrow[0,1]\) be a distribution function.

**a)** Show that there exists a probability space \((\Omega, \mathscr{A}, \mathbb{P})\) and a random variable \(X\) such that \(F(x)=\mathbb{P}(X \leqslant x)\).

**b)** Show that there exists a probability space \((\Omega, \mathscr{A}, \mathbb{P})\) and an iid sequence of random variables \(X_{n}\) such that \(F(x)=\mathbb{P}\left(X_{n} \leqslant xight)\).

**c)** State and prove the corresponding assertions for the \(d\)-dimensional case.

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**Related Book For**

## Brownian Motion A Guide To Random Processes And Stochastic Calculus De Gruyter Textbook

**ISBN:** 9783110741254

3rd Edition

**Authors:** René L. Schilling, Björn Böttcher