What is a Geometric Distribution?

A Geometric Distribution is a discrete Random Sampling Distribution. The sampling is from a series of independent trials each of which may have outcomes which fall into one of two classifications. The Geometric Distribution generates the probability of X-trials required until the first occurrence is obtained. The probability of occurrence of each classification remains constant for each trial. The outcomes of each successive trial also remains independent. This sampling situation is sometimes referred to as a Bernoulli Trial or Process.

Calculations

The probability of exactly X trials required to obtain the first occurrence may be calculated by the following formula.

Geometric Distribution Calculation

Where:

  • X is the number of Trials, and
  • Pi is the probability of occurrence.

Other Properties

The Geometric Distribution has a Mean and a standard deviation as follows.

Geometric Distribution Mean and Sigma

Example

Here is an example Geometric Distribution, assuming a probability of occurrence of 0.5. The probability of 8 trials before an occurrence is observed is 0.0039, as seen in the table below.

Geometric Distribution

      p = 0.5    E(x)=2.00

    Prob       Equal &     Equal &

   r      at r        Above       Below
   1      0.5000      1.0000      0.5000 |-----------------------------------
   2      0.2500      0.5000      0.7500 |------------------
   3      0.1250      0.2500      0.8750 |---------
   4      0.0625      0.1250      0.9375 |----
   5      0.0313      0.0625      0.9688 |--
   6      0.0156      0.0313      0.9844 |-
   7      0.0078      0.0156      0.9922 |-
   8      0.0039      0.0078      0.9961 |
   9      0.0020      0.0039      0.9980 |
  10      0.0010      0.0020      0.9990 |
  11      0.0005      0.0010      0.9995 |
  12      0.0002      0.0005      0.9998 |
  13      0.0001      0.0002      0.9999 |
  14      0.0001      0.0001      0.9999 |
  15      0.0000      0.0001      1.0000 |
  16      0.0000      0.0000      1.0000 |
   \          \           \           \
 Inf.     0.0000      0.0000      1.0000 |