714. Geometric Random Variable - JulTob/Mathematics GitHub Wiki

Definition

A geometric random variable is based on an experiment consisting of repeating a Bernoulli trial and recording the repetition in which the first success occurs. Therefore, the minimum value of this variable is one. Intuitively, as the experiment is repeated, the probability associated with a success at a later repetition decreases rapidly unless the probability of success is low.

The geometric random variable is characterized solely by the probability of success pp in the associated Bernoulli trial.

Parameter

Probability of success: p=0.4p = 0.4

Probability Function

Each value has a specific probability of occurrence:

Variable value kk	Probability pX(k)p_X(k)	kpX(k)k p_X(k)	k2pX(k)k^2 p_X(k)
1	0.4	0.4	0.4
2	0.24	0.48	0.96
3	0.144	0.432	1.296
5	0.05184	0.2592	1.296
6	0.031104	0.186624	1.119744
7	0.0186624	0.1306368	0.9144576
8	0.01119744	0.08957952	0.71663616
9	0.006718464	0.060466176	0.544195584
10	0.0040310784	0.040310784	0.40310784
11	0.00241864704	0.02660511744	0.2926562918
12	0.001451188224	0.01741425869	0.2089711043
13	0.0008707129344	0.01131926815	0.1471504859

Properties

The variable represents the trial number at which the first success occurs.
The probability of obtaining a success on later trials decreases exponentially.
The expectation is given by 1p\frac{1}{p}.
The variance is given by 1−pp2\frac{1-p}{p^2}.

Applications

This type of random variable is widely used in:

Modeling the number of attempts before success, such as in gambling or manufacturing processes.
Reliability analysis, where it helps predict the number of trials until a failure occurs.
Queueing theory, where it models the waiting time until an event happens.

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    title "L=10"
    x-axis 0 --> 15
    y-axis "Chances" 0 --> 1
    bar [ 0.6666666667, 0.4, 0.24, 0.144, 0.05184, 0.031104, 0.0186624, 0.01119744, 0.006718464, 0.0040310784, 0.00241864704, 0.001451188224, 0.0008707129344 ]
    line [ 0, 0.4, 0.48, 0.432, 0.2592, 0.186624, 0.1306368, 0.08957952, 0.060466176, 0.040310784, 0.02660511744, 0.01741425869, 0.01131926815 ]