I want to proof that If$$X∼N(0,1)$$, then$$Y=μ+σX$$has the normal distribution with mean $μ$ and variance $σ^2$. I searched it before, but I don't understand why I have to calculate the probability density of X and also Jacobian.

Can someone explain me clearly? Thank you in advance.

  • $\begingroup$ Considering CDFs or CFs suffice. $\endgroup$ – Nap D. Lover Apr 17 at 19:03

Characteristics of normal distribution:

  • If random variable $X$ has normal distribution and $Y=a+bX$ where $a,b$ are constants and $b\neq0$ then also $Y$ has normal distribution.
  • The distribution is completely determined by mean and variance.

According to first bullet $Y=\mu+\sigma X$ has normal distribution if $X\sim\text{Norm}(0,1)$ with: $$\mu_Y=\mathbb E(\mu+\sigma X)=\mu+\sigma\mathbb EX=\mu\text{ and }\sigma_Y^2=\mathsf{Var}(Y)=\mathsf{Var}(\mu+\sigma X)=\sigma^2\mathsf{Var}(X)=\sigma^2$$This justifies the conclusion that $Y\sim\text{Norm}(\mu,\sigma^2)$.

It might be that actually you want a proof of the first bullet.

If $\Phi$ denotes the CDF of $X$ and $\phi$ the PDF of $X$ then: $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac12x^2}=\Phi'(x)$$

We can find CDF $F_{Y}(y)=P(\mu+\sigma X\leq y)=P(X\leq\frac{y-\mu}{\sigma})=\Phi(\frac{y-\mu}{\sigma})$ and - taking the derivative - PDF: $$f_Y(y)=\frac1{\sigma}\phi\left(\frac{y-\mu}{\sigma}\right)=\frac1{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{y-\mu}{\sigma}\right)^2}$$

which is the PDF associated with $\text{Norm}(\mu,\sigma^2)$.

  • $\begingroup$ why 𝜎𝔼𝑋 is disappearing in 𝜇+𝜎𝔼𝑋=𝜇 ? $\endgroup$ – RecklessSerenade Apr 19 at 2:46
  • 1
    $\begingroup$ Because $\mathbb EX=0$ if $X\sim\text{Norm}(0,1)$. $\endgroup$ – drhab Apr 19 at 7:25


$$\mathbb P\{Y\leq y\}=\mathbb P\left\{X\leq \frac{y-\mu}{\sigma }\right\}.$$


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