Geometric Distribution

This is a special case of Negative Binomial Distribution. Here we only consider the failures till we get the first success.

So this is Negative Binomial Distribution with $r=1$

1. The Core Idea

Imagine you are trying to roll a 6 on a standard die. You roll again and again until you finally see a 6, and then you stop instantly.

• Success ($1$) happens with probability $p$
• Failure ($0$) happens with probability $q = 1 - p$

If all trials are $X$, we will have $x-1$ times of failures before getting the success.

$$X \sim \text{Geometric}(p)$$

$X$ - Number of Trials till we get the first success

PMF

$$P(X = x) = (1 - p)^{x - 1} p$$

Properties

Expected Value:

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

If success is $10\%$, in average it takes $\frac{1}{0.1}=10$ trials in average.

Variance:

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

Memoryless!

• Only discrete probability distribution that has Memorylessness of Distributions property.

Why called geometric?

This comes from the geometric series in mathematics. Not any circles or triangles.

A geometric sequence is something which has each next term multiplied by a common factor $r$

$$S= \{a, \; ar, \; ar^2, \; ar^3, \; ar^4, \dots \}$$

• The first term is $a = p$
• The common ratio is $r = (1 - p)$

Because total probability must equal $1$, summing these probabilities relies directly on the classic infinite geometric series formula:

$$S_n = \frac{a_1(1 - r^n)}{1 - r}$$ $$\sum_{y=1}^{\infty} P(Y = y) = p + p(1-p) + p(1-p)^2 + \dots = \frac{p}{1 - (1 - p)} = \frac{p}{p} = 1$$