B2 - common discrete random variables

a discrete RV is associated with a discrete probability distribution

distribution functions

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

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

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

the cumulative distribution function (CDF) of $X$ is defined as:

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

the tail of $X$ is defined as:

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

bernoulli ( $p$ )

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}

$X$ is said to be a RV drawn from the bernoulli ( $p$ ) distribution

X \sim Bernoulli (p)

the PMF:

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

binomial $(n, p)$

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

$X$ is said to be a RV drawn from the binomial ( $n, p$ ) distribution:

X \sim Binomial (n, p)

geometric ( $p$ )

again, considering the coin flip, but now done until a head is obtained
these are independent trials, each distributed $Bernoulli (p)$
let RV $X$ represent the number of flips until success (heads)
the PMF is defined as:

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

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

$X$ is said to be a RV drawn from the geometric ( $p$ ) distribution:

X \sim Geometric (n, p)

poisson ( $λ$ )

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