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## Sunday, July 7, 2013

### geometric distribution - basics illustrated

This is prerequisite of Geometric Brownian Motion.., but this is a Distribution whereas GBM is a Process.

Based on http://en.wikipedia.org/wiki/Geometric_distribution --

Keep rolling a dice until you get a 6. Record the number of rolls required (X) in a log book, and repeat the game to get another X, and another X.... X follows a geometric distribution. Discrete, so histogram shows --
Height of Bar#1 is 1/6
Height of Bar#2 is 1/6 x 5/6 i.e. the likelihood of X = 2
Height of Bar#3 is 1/6 x 5/6 x 5/6
...
Height of Bar#99 is 5^98 / 6^99 a very small number.
....
Each bar is shorter than the previous bar, i.e. 83.3%
Total lengths must add up to 1.0

This histogram completely describes the distribution of the discrete random variable X. From this histogram we can derive the Expected, variance, stdev etc.

Not intuitive (better internalize), but E(X) = 1/p and is 6 in our case.
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Many everyday probability problems involve GD. Many quant interview questions involve some GD but sometimes much harder than GD. It pays to have a firm grounding on the basics of GD.

Why call it "geometric"? Because each bar is 83.3% the height before. If you space out the bars equally, it's a gentle down-slope.