Distribution

13 Feb 2018 on Math

1. continuous probability Distribution

probability distribution that has a cumulative distribution function that is continuous

- normal distribution(= Gaussian normal distribution)

$\mathcal{N}(x; \mu, \sigma^2) = \dfrac{1}{\sqrt{2\pi\sigma^2}} \exp \left(-\dfrac{(x-\mu)^2}{2\sigma^2}\right)$

- standard normal distribution

when ${\displaystyle \mu =0}$ and ${\displaystyle \sigma =1}$

$\mathcal{N}(x; \0, \1) = \dfrac{1}{\sqrt{2\pi}} \exp \left(-\dfrac{x^2}{2}\right)$

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- central limit Theorem

when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed

$S_{n}:={\frac {X_{1}+\cdots +X_{n}}{n}}$

${\sqrt {n}}\left(\left({\frac {1}{n}}\sum _{i=1}^{n}X_{i}\right)-\mu \right)\ {\xrightarrow {d}}\ N\left(0,\sigma ^{2}\right)$

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- Student t distribution

any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown

$t(x;\mu, \sigma^2, \nu) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)} {\sqrt{\nu\pi}\Gamma\left(\frac{\nu}{2}\right)} \left(1+\frac{(x-\mu)^2}{\nu\sigma^2} \right)^{-\frac{\nu+1}{2}}$