Geometric Distribution

A Discrete Distribution that can lead to Countably Infinite sequence of Independent Bernoulli Trials.

RV $X/Y$ defined as $\text{\# of failures/trials before 1st success}$

• $Y=X+1$

Commonly used when:

• Finding the waiting time of something to occur It is Memorylessness

PMF

• $p$ = probability of success.
• $q$ = probability of failure
• $x=$ random variable for number of failures before success occurs. usually its success - 1.
• $y=$ number of trials $P(X=x)=q^{x}p,\forall x\in {\mathbb{N}}_0$ $P(Y=y)=q^{y-1}p,\forall y\in \mathbb{N}$

CDF

$F_X(x)=P(X<x)={\sum }_{k=0}^x{p}_x(k)=1-q^{x+1},\forall x\in \mathbb{N}$

Expectation formula

$E(Y)=\sum _{y=1}^{\infty }=\frac{q}{p}$

Example

Calculate the probability distribution for getting out of jail in MONOPOLY in x rolls of dice.

• $p=6/36=1/6$
• $q=5/6$
• $E(x)=(5/6)/(1/6)=5$
• Thus, we have 5 tries before getting a success

Geometric Distribution Table

With p = 0.1 q = 0.9

x	P(x)
0	0.9^0 * 0.1
1	0.9^1 * 0.1 = 0.09
2	0.9^2 * 0.1 = 0.081
3	0.9^3 * 0.1 = 0.0729