Geometric Distribution#
TODO
Geometric Random Variable#
Definition#
TODO
Geometric Conditions#
TODO
TODO
Geometric Parameters#
A Geometric Random Variable has single parameter.
Geometric Probability#
TODO
Probability Density Function#
TODO
\[P(\mathcal{X} = x) = \sum_{i=1}^{x} (1-p)^{x-1} \cdot p\]
TODO
Cumulative Distribution Function#
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In order to show the geometric density represents a distribution, we must show the probability of all outcomes sums to 1. In order to do this, we must first talk about the geometric series.
Geometric Series#
A geometric series is defined as the sum of powers of r,
\[\sum_{i=1}^{n} r^i = r + r^2 + r^3 + ... + r^n\]
The reason it is called geometric can be easily seen if we give r a value. For instance, if \(r = \frac{1}{4}\), then the first few terms of the geometric series are given by,
\[\sum_{i=1}^{n} (\frac{1}{4})^i = \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + ... + (\frac{1}{4})^n\]
Each term on the right hand side can be identified with the areas of successive squares in the following picture,
TODO
Expectation#
TODO