Give probability space $(\Omega,F,\mathbb P)$ & random variable $X:\Omega \to \mathbb R$ on $(\Omega,F,\mathbb P)$ so $X$ has normal distribution.

Question

Problem

A random variable $X$ is said to have a standard normal distribution if $X$ is absolutely continuous with density given by $$\frac{d\mathbb P_X}{d \lambda_1}(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac 12x^2}, \qquad x\in \mathbb R $$

Construct [Give an example of] a probability space $(\Omega,F,\mathbb P)$ and a random variable $X: \Omega \to \mathbb R$ on $(\Omega,F,\mathbb P)$ such that $X$ has a standard normal distribution. In your example, be sure to verify that $X$ does indeed have a standard normal distribution.

I have no idea on how to construct this kind of probability space based on the given distribution. Please guide.

Answer 1

I will guide you through this exercise. As my professor used to tell us, make it as easy as posible for yourself by taking $X(\omega)=\omega$ in an appropriate space. So let us take $\Omega=\mathbb R$ and $\mathcal F=\mathcal B$ the Borel-$\sigma$-algebra. We want to have $$\mathbb P(X\leq x)=\mathbb P(\{\omega \in\mathbb R\ : \ \omega\leq x\}) $$ to be equal to $$\int^x_{-\infty} \frac{1}{\sqrt{2\pi}}e^{-t^2/2}\,dt$$ So we take $$\mathbb P(A)=\int_A \frac{1}{\sqrt{2\pi}}e^{-t^2/2}\,dt $$ for every Borel measurable set $A$. Is this $\mathbb P$ a probability measure? Is our $X$ measurable? Does it have the CDF that we want?