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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.

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closed as off-topic by Did, Davide Giraudo, KReiser, user10354138, Cesareo Nov 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 $\Phi$.

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