## Sunday, September 19, 2010

### 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.

$p_X(k)$ = $\frac{1}{b-a+1}$ if k = a, a+1, a+2, ..., b-1, b
$p_X(k)$ = 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(k)$= p if k = 1$p_X(k)$= 1-p if k = 0 E[X] = p var(X) = p(1-p) Binomial with Parameters p and n It describes the number of successes in n independent bernoulli trials. For example, total number of HEADs obtained in n tosses.$p_X(k)$=${n \choose k}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(k)$=$(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

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

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