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:
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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.

The probability mass function of a geometric distribution with parameter p forms a geometric progression

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
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