<< probability_theory

Continuous Distribution

1. Normal Distribution

For the random variable with probability density function $$p(x)=\frac{1}{\sqrt{2\pi }\sigma }\exp \{-\frac{(x-\mu {)}^2}{2{\sigma }^2}\}$$ where $\mu ,\sigma >0$ are parameters, we define $X$ complies the normal distribution $X\sim N(\mu ,{\sigma }^2)$.

Regularity Checking.

Let ${\displaystyle x^{\prime }=\frac{x-\mu }{\sigma }}$, $$F(+\mathrm{\infty })={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}p(x)\text{d}x={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\sqrt{2\pi }\sigma }\exp \{-\frac{(x-\mu {)}^2}{2{\sigma }^2}\}\text{d}x={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\phi (x)=\frac{1}{\sqrt{2\pi }}{\text{e}}^{-\frac{x^{\prime 2}}{2}}$$ Let $x^{\prime }=\sqrt{2t}$, $$F(+\mathrm{\infty })=$$

Specially, when $\mu =0,\sigma =1$, the normal distribution is called standard normal distribution, denoted as $X\sim N(0,1)$. The probability density function of standard normal distribution is denoted as $$\phi (x)=\frac{1}{\sqrt{2\pi }}{\text{e}}^{-\frac{x^2}{2}}$$ and its cumulative distribution function denoted as $\mathrm{\Phi }(x)$. Notice that $\phi (x)$ is even function hence $\mathrm{\Phi }(x)+\mathrm{\Phi }(-x)=1$.

Any normal distribution $X\sim (\mu ,{\sigma }^2)$ can be converted to the standard form by linear transformation: $$Y=\frac{X-\mu }{\sigma }\sim N(0,1)$$

Proof. The cumulative distribution function of constructed random variable $Y$ is $$F_Y(y)=P(\frac{X-\mu }{\sigma }\le y)=P(X\le \sigma y+\mu )={\int }_0^{\sigma y+\mu }\frac{1}{\sqrt{2\pi }\sigma }\exp \{-\frac{(x-\mu {)}^2}{2{\sigma }^2}\}$$ Let $x=\sigma y+\mu$, we have $$F_Y(y)={\int }_0^y\frac{1}{\sqrt{2\pi }\sigma }{\text{e}}^{-\frac{y^2}{2}}$$ and the probability density function $$p_Y(y)=\frac{\text{d}{F}_Y(y)}{\text{d}y}=\frac{\text{d}\left({\displaystyle {\int }_0^y\frac{1}{\sqrt{2\pi }\sigma }{\text{e}}^{-\frac{y^2}{2}}}\right)}{\text{d}y}=\frac{1}{\sqrt{2\pi }\sigma }{\text{e}}^{-\frac{y^2}{2}}\sim N(0,1)$$

The most simple form of normal distribution is

2. Uniform Distribution

The uniform distribution is random variable with probability density function $$p(x)=\{\begin{array}{ll}\frac{1}{b-a}& a\le x\le b\\ 0,& x<a,x>b\end{array}$$ denoted as $X\sim U(a,b)$.

3.