Continuous Normal Distribution

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A kind of Continuous Distribution that is symmetric and bell-shaped, completely characterized by its mean $\mu$ and variance ${\sigma }^2$, where outcomes cluster around the center.

$$X\sim \text{N}(\mu ,{\sigma }^2)\text{for }x\in \mathbb{R},{\sigma }^2>0$$

• $f(x)=\frac{1}{\sigma \sqrt{2\pi }}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right)\text{for }x\in (-\infty ,\infty )$
• $E(X)=\mu$
• $\text{Var}(X)={\sigma }^2$
• There is no analytic solution for: $F_X(x)={\int }_{-\infty }^x\frac{1}{\sigma \sqrt{2\pi }}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right)$

Standard Normal Distribution

Continuous Standard Normal Distribution

A Continuous Normal Distribution that is transformed as follows.

$$Z\sim \text{N}(0,1)\text{where }\mu =0,{\sigma }^2=1$$

• PDF: $\varphi (z)=\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{z^2}{2}\right)$
• $E(Z)=0$
• $\text{Var}(Z)=1$
• CDF: $\Phi (z)={\int }_0^x\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{z^2}{2}\right)dx$
  • No analytic solution. Use a Z Table.

Any normal distribution can be easily changed to a standard normal distribution

$$Z=\frac{X-\mu }{\sigma }\sim \text{N}(0,1)$$

• $F(Z)=\Phi (z)=P(Z\le z)={\int }_{-\infty }^z\varphi (t)dt={\int }_{-\infty }^z\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{t^2}{2}dt\right)$

Properties

• $\varphi (x)=\varphi (-x)$
• $\Phi (-x)=1-\Phi (x)$
• $P(Z>z)=1-P(Z<-z)$
• $P(a<Z<b)=P(Z<b)-P(Z<a)=\Phi (b)-\Phi (a)$
• If $P(Z<z_{\alpha })=\alpha$, then $P(Z>z_{\alpha })=1-\alpha$
• If $\Phi (z_{\alpha })=\alpha$, then $z_{\alpha }={\Phi }^{-1}(\alpha )$
• $z_{1-\alpha }=-z_{\alpha }$

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