Appendix A — Random Variable Summary

A.1 RV summary

\(X\)	Binomial	Geometric	Poisson	Discrete Uniform	Normal	(Continuous) Uniform	Exponential
Type	Discrete	Discrete	Discrete	Discrete	Continuous	Continuous	Continuous
Parameters	\(n\), \(p\)	\(p\)	\(\lambda\)	\(a\), \(b\)	\(\mu\), \(\sigma^2\)	\(a\), \(b\)	\(\lambda\)
Description	Number of successes in \(n\) independent trials with \(p\) probability of success for each trial (Note: Bernoulli is just Binomial with \(n\!=\!1\))	Number of failures BEFORE the first success while independently repeating trial with \(p\) probability of success	Count of number of occurrences of an event with constant mean rate \(\lambda\) that’s independent of previous occurrences	\(n\)-sided fair die	Normal distributions usually arise from CLT (i.e. they’re processes that are the sum of many smaller independent processes)	Generalizing \(n\)-sided fair die to a continuous interval	Waiting time between Poisson events
Outcomes	\(0,1,\ldots,n\)	\(0,1,\ldots\)	\(0,1,\ldots\)	\(a,a\!+\!1,\ldots,b\)	\((-\infty,\infty)\)	\([a,b]\)	\([0,\infty)\)
PDF/PMF at \(k\)	\({n\choose k}p^k(n-p)^{n-k}\)	\(p(1-p)^k\)	\(\frac{\lambda^ke^{-\lambda}}{k!}\)	\(\frac1{b-(a-1)}\)	\(\frac1{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}\sigma\right)^2}\)	\(\frac1{b-a}\)	\(\lambda e^{-\lambda x}\)
\(P(X\le k)\)		\(1-(1-p)^{\lfloor k\rfloor+1}\)		\(\frac{\lfloor k\rfloor-(a-1)}{b-(a-1)}\)		\(\frac{x-a}{b-a}\)	\(1-e^{-\lambda x}\)
Mean	\(np\)	\(\frac{1-p}p\)	\(\lambda\)	\(\frac{a+b}2\)	\(\mu\)	\(\frac{a+b}2\)	\(\frac1\lambda\)
Variance	\(np(1-p)\)	\(\frac{1-p}{p^2}\)	\(\lambda\)	\(\frac{(b-(a-1))^2-1}{12}\)	\(\sigma^2\)	\(\frac{(b-a)^2}{12}\)	\(\frac1{\lambda^2}\)
R functions	dbinom, pbinom, qbinom, rbinom	dgeom, pgeom, qgeom, rgeom	dpois, ppois, qpois, rpois	sample	dnorm, pnorm, qnorm, rnorm	dunif, punif, qunif, runif	dexp, pexp, qexp, rexp