Some Discrete Random Variables

From Introduction to Probability.


Discrete Uniform over [a,b]
When X can take any value from a to b(both inclusive) with equal probability.

$p_X\left(k\right)$ = $\frac{1}{b-a+1}$ if k = a, a+1, a+2, ..., b-1, b
$p_X\left(k\right)$ = 0 otherwise

E[X] = $\frac{a+b}{2}$

var(X) = $\frac{(b-a)(b-a+2)}{12}$


Bernoulli with Parameter p
It describes the success or failure in a single trial. For example, we can say that X = 1 if a toss results in HEAD or else X = 0 if the toss results in TAIL and probability of HEAD is p.

$p_X\left(k\right)$ = p if k = 1
$p_X\left(k\right)$ = 1-p if k = 0

E[X] = p

var(X) = p(1-p)


Binomial with Parameters p and n
It describes the number of successes in n independent bernoulli trials. For example, total number of HEADs obtained in n tosses.

$p_X\left(k\right)$ = $\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$, if k = 0,1,2,...n

E[X] = np

var(X) = np(1-p)

Also, we can use bernoulli random variables to model binomial random variable as follow.
Let $X_1,X_2,....X_n$ are n independent Bernoulli Random variables with parameter p. Then
Binomial Random Variable, X = $X_1+X_2+...+X_n$


Geometric with Parameter p
It describes the number of trials until the first success, in a sequence of independent Bernoulli trials. For example, the number of tosses required to get first HEAD.

$p_X\left(k\right)$ = $(1-p{)}^{k-1}p$, if k = 1,2,....

E[X]] = $\frac{1}{p}$

var(X) = $\frac{1-p}{p^2}$


Poisson with Parameter $\lambda$
It approximates the binomial PMF when n is large and p is small and $\lambda$ = np

$p_X\left(k\right)$ = $e^{-\lambda }\frac{{\lambda }^k}{k!}$, k = 0,1,2...

E[X] = $\lambda$ = var(X)