Derivations for moments of univariate normal distribution#

Standard Normal#

Expectation#

\[\begin{split} \begin{array}{c} X{\sim} p(x) \\ \sim \mathcal{N}(0,1) \\ =\frac{1}{\sqrt{2 \pi}}e^{(-1 / 2) x^{2}} \\ E_{X}[x]=\int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}}e^{(-1 / 2) x^{2}} \cdot x dx \end{array} \end{split}\]

Solve indefinite integral, $\( \begin{array}{l} I=\int \frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} \cdot x d x \\ y=\frac{-x^{2}}{2} \Rightarrow d y=-x d x \\ \therefore I=\int \frac{-1}{\sqrt{2 \pi}} e^{y} d y=\frac{-1}{\sqrt{2 \pi}} \int e^{y} d y=\frac{-1}{\sqrt{2 \pi}} e^{y} \end{array} \)\( Substitute back \)\( I=\frac{-1}{\sqrt{2 \pi}} e^{-x^{2} / 2} \)\( Definite integral is \)\left.\frac{-1}{\sqrt{2 \pi}} e^{-x^{2}}\right|_{-\infty} ^{\infty}=\frac{-1}{\sqrt{2 \pi}}\left(e^{-\infty}-e^{-\infty}\right)=0$

Variance#

Affine Transformation of Standard Normal#

References

  1. https://www.youtube.com/watch?v=U-13q74uvTo

  2. https://www.youtube.com/watch?v=mrcRODAx-Vk

  3. https://en.wikipedia.org/wiki/Gaussian_integral

  4. https://www.youtube.com/watch?v=zbHRUnR9F-4 (Very detailed!)

  5. https://www.youtube.com/watch?v=l27xKSNad2Y (Very intuitive double integral explanation)