Home page for accesible maths 5 Models for discrete random variables 5.8 Hypergeometric random variables (not examinable)6 Continuous random variables

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

5.9 Summary

There are many discrete probability models based on Bernoulli trials (eg coin tossing). The basics are:

•

the sample space ${\Omega=\{\omega|\mbox{seq of H,Ts}\}}$ ,
•

the induced sample space ${{\mathcal{S}}}$ , usually ${\{0,1,2,\dots\}}$ ,
•

independent trials, with constant probability on each trial ${{\rm P}(H)=\theta}$ .

A random variable ${R}$ is function from ${\Omega}$ to induced ${{\mathcal{S}}}$ , with pmf ${p(r)={\rm P}(\{R=r\})}$ . The construction of ${R}$ determines if the pmf is Bernoulli, Binomial, Geometric etc:

	${{\mathcal{S}}}$	construction	${p_{R}(0)}$
Discrete uniform	$\{0,1,\ldots,n\}$	Dice roll	$(n+1)^{-1}$
Bernoulli	${\{0,1\}}$	single throw	${1-\theta}$
Binomial	${\{0,1,\dots,n\}}$	# Hs in ${n}$ throws	${(1-\theta)^{n}}$
Geometric	${\{0,1,\dots\}}$	# Ts before H	${\theta}$
Poisson	${\{0,1,\dots\}}$	Bino limit ${n\theta{\rightarrow}\lambda}$	${\exp(-\lambda)}$