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Generating Geometric distributions using induction

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I need assistance with the attached problem. It requires me to show that a given algorithm generates the geometric distribution.

Please see the attached document for details.

Show that the following algorithm is valid for generating X -- geom(p)

1. Ler i=0.
2. Generate (please see the attached file) independent of any previously generated U(0,1) random variates.
3. If U is lesser than or equal to p, return X = i. Otherwise, replace i by i + 1 and go back to step 2.

Note:
U(0,1) represents the uniform distribution where 0 is lesser than or equal to u, which is lesser than or equal to 1.

Geom(p) represents the geometric distribution.
f(x) = p(1 -p)^x, x = 0,1,2 ... is the pdf of the geometric distribution with mean (1-p)lp

Please show all your work.
Thank you.

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Solution Summary

This solution explains how to generate geometric distributions using induction.

Solution Preview

This can be shown by induction.

x = 0.

The geometric distribution tells us that f(0) = p. We need to verify that the probability of x = 0 is also p using the algorithm.

This is simple because i is increasing each time, we either get 0 on the first run or we will never get zero again (and this is true with every number). The probability of getting zero on the first run is ...

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