B2 - common discrete random variables

distribution functions

let $X$ be a discrete RV, then the probability mass function (PMF) is defined as:

p_{X} (a) = P (X = a))

where, $\sum_{x} p_{X} (x = 1$

F_{X (a)} := P {X \leq a} \sum_{x \leq a} p_{X} (c)

{\bar{F}}_{X} (a) = P {X > a} = \sum_{x > a} p_{X} (x) = 1 - F_{X} (a)

use case	distribution	PMF
one trial, two outcomes	bernoulli	$p_{X} (1) = p$ $p_{X} (0) = 1 - p$
fixed number of trials	binomial	$p_{X} (i) = P (X = i} = (\binom{n}{i}) p^{i} (1 - p)^{n - i}$
until first success	geometric	$p_{X} (i) = P (X = i) = (1 - p)^{i - 1} p$
counting events over time/space	poisson	$p_{X} (i) = P (X = i) = e^{- λ} \frac{λ^{i}}{i!}$

bernoulli ( $p$ )

used when there are only two possible outcomes
considering an experiment with a single coin flip, where the probability of heads is $p$ , hence tails is $1 - p$
let RV $X$ represent the outcome, ie. coin value, so $1$ is heads and $0$ otherwise

X = {\begin{cases} 1 & with probability p, \\ 0 & otherwise. \end{cases}

X \sim Bernoulli (p)

\begin{aligned} p_{X} (1) & = p \\ p_{X} (0) & = 1 - p \end{aligned}

used when a two-outcome experiment is repeated a fixed number of times, and the number of successes is being counted
now considering that the coin flip is done $n$ times (independently)
let $X$ represent the number of heads, ie: $X = {0, 1, 2, \dots, n}$

p_{X (i)} = P (X = i) = (\binom{n}{p}) p^{i} (1 - p)^{n - i}

where, $i = 0, 1, 2, \dots, n$

X \sim Binomial (n, p)

p_{X} = P {X = i} = (1 - p)^{i - 1} p

where, $i = 1, 2, 3, \dots$

X \sim Geometric (n, p)

used for the number of events in a fixed interval of time, space, or area, for events happening independently at an average rate
eg: number of calls a helpline receives in an hour
the $Poisson (λ)$ distribution is defined via its PMF, and arises naturally when looking at a mixture of a very large number of independent sources, each with a very small individual property
if $X \sim Poisson (λ) :$

p_{X} (i) = \frac{e^{- λ} λ^{i}}{i!}

where, $i = 0, 1, 2, \dots$

note that the plot for the poisson distribution, like that for the binomial, has a bell-like shape, but with an infinite range