Prob Summary

Distributions

Either discrete or continuous or mixed.

Discrete Distributions

given with the probability function $P(X=x)$, where $\sum _{x}P(X=x)=1$.

$$\mu =E(X)=\sum _{x}x\cdot P(X=x)$$$${\sigma }^2=var(X)=E(X-\mu {)}^2,\sigma =\sqrt{var(X)}$$

Continuous Distributions

Continuous distributions are given are given by the density function f(x), such that $P(a<x<b)={\int }_a^{b}f(x)dx$

$$\mu =E(X)={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}x\cdot f(x)$$$$var(X)=E(X^2)-(EX{)}^2\text{(while }E(X^2)={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{x}^{2}f(x)dx$$

Discrete Distributions

Poisson Distribution

x counts the number of "rare events" in an area and/or period, with expectation $\mu$

$$P(X=x)=\frac{{\mu }^x}{x!}{e}^{-\mu }$$$$E(X)=var(X)=\mu$$

Hypergeometric Distribution

n draws without replacement from R red and N-R white balls; X = # of red balls.

$$P(X=x)=\frac{(\genfrac{}{}{0ex}{}{R}{x})(\genfrac{}{}{0ex}{}{N-R}{n-x})}{(\genfrac{}{}{0ex}{}{N}{R})}$$

Binomial Distribution

applies to situations where we count the number of successes in n Bernoulli trials with success rate p: X = # of successes

$$P(X=x)=(\genfrac{}{}{0ex}{}{n}{x})p^x(1-p{)}^{n-x}$$$$E(X)=np$$$$var(X)=np(1-p)$$

Normal approx of the binomial Distribution

with the $N(np,np(1-p))-dist$ if $n\ge 25,np\ge 5\text{ and }n(1-p)\ge 5$

Don't forget the continuity correction.

Continuous Distributions

Uniform Distribution

Model for random numbers drawn from an interval, Especially (0, 1).

$$UU(a,b)$$$$f(x)=\frac{1}{b-a},a\le x\le b$$$$E(X)=\frac{a+b}{2}$$$$var(X)=\frac{(b-a{)}^2}{12}$$

Exponential Distribution

Model for waiting times, inter-arrival times and lifetimes

$$UExp(\lambda )$$$$f(x)=\lambda e^{\lambda x},x\ge 0$$$$E(X)=\frac{1}{\lambda }$$$$var(X)=\frac{1}{{\lambda }^2}$$$$P(X>x)=e^{-\lambda x}$$

Normal Distribution

Model for “natural quantities” variables in nature, economy, etc.

$$XN(\mu ,{\sigma }^2$$$$E(X)=\mu$$$$var(X)={\sigma }^2$$$$Z=\frac{X-\mu }{\sigma }N(0,1)$$