# Prove that random variable has standard normal distribution [closed]

How do I prove that random variable X has a standard normal distribution given the probability density function?

A random variable X has a standard normal distribution if X is absolutely continuous with density given by: $$\frac{d\mathbb{P}_X}{d\lambda_1}(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2} x^2},\: x\in\mathbb{R}.$$

Provide example of probability space $$(\Omega, \mathcal{F}, \mathbb{P})$$ and a random variable $$X:\Omega\to\mathbb{R}$$ on $$(\Omega, \mathcal{F}, \mathbb{P})$$ and verify that $$X$$ has a standard normal distribution.

## closed as off-topic by Did, Davide Giraudo, KReiser, user10354138, CesareoNov 21 '18 at 2:16

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Let $$\Phi$$ denote the standard normal distribution. Let $$\Omega =(0,1),\mathcal F =$$ Borel sigma algebra and let $$P$$ be the Lebesgue measure. Let $$X(\omega)=\Phi ^{-1} (\omega)$$. ($$X$$ is a random variable because it is a continuous function on $$(0,1))$$. We have $$Pr\{ X\leq t\}=P\{\omega: \Phi ^{-1} (\omega) \leq t\}=P\{\omega: \omega \leq \Phi (t)\}=\Phi (t)$$ so $$X$$ has distribution $$\Phi$$.