# Geometric Distribution

A discrete random variable is said to follow a geometric distribution with parameter p written X~Ge(p)

if its probability distribution satisfies the following equation:

SS

### Properties

Expected Value E(x) = 1 / (1-p)

Variance V(x) = P / (1-p)2

### Assumptions

1) The total number of trials is potentially infinite

2) There are only 2 outcomes, success or failure

3) The outcomes are independent

4) All the trials have the same probability of success

### Related to Binomial Distribution

Both these distributions are based on independent trials where the probability of success is constant.

A geometric random variable is the number of trials until the first failure.

A binomial random variable is the number of success in "n" trials.

### Geometric Series

A geometric series is a sum of either a finite or an infinite number of terms.

Each term after the first term of a geometric series is a multiple of the previous term by some fixed constant, x.

Multiplication of a geometric series by a constant does not affect its nature.

### The Finite Geometric Series

An example would be: 25 + 50 + 100 + 200 + 400 is a geometric series because each term is twice the previous term.

The most basic geometric series is 1 + x + x2 + x3 + x4 + ... + xn.

This is the finite geometric series because it has exactly n + 1

Solve 1 + 10 + 102 + 103 + ... + 10

Solve 1 - 3 + 9 - 27 + ... + (-3)10

1/2 + 3 - 18 + 108 - ... - 23328

### Infinite Geometric Series

An example would be: 4 + 2 + 1 + .5 + .25 + .125 + .625 + ... is an (infinite) geometric series because each term is 1/2 the previous term.

The most basic geometric series is 1 + x + x2 + x3 + x4 + …

There are 2 variations of infinite geometric series

SS

© 2023 Better Solutions Limited. All Rights Reserved. © 2023 Better Solutions Limited TopPrevNext