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February 9th, 2014, 06:43 AM   #1
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Homework help - Generating random variable

Hi, I have a problem on the following homework assignment:

Suppose that X and Y are discrete random variables with a joint probability
mass function p_XY (x, y).

Procedure is the following:
a. Generate X ? p_X (x).
b. Accept X with probability p(y|X).
c. If X is accepted, terminate and return X. Otherwise go to Step a.

Show that this procedure generates a random variable X ? p_X|Y (x|y).


I would've been grateful for any help. Tnx
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February 9th, 2014, 11:11 AM   #2
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Re: Homework help - Generating random variable

I am assuming that by "generate some random variable" you mean to generate random outcomes that follow that particular distribution.
The idea is to generate first random outcomes from a Uniform Random Variable in (0,1) for which there are plenty libraries. For instance, in java you would do something like:

double uniformOutcome;
Random rand;
uniformOutcome = rand.nextDouble();

The generated number is a "pseudorandom" meaning it is not really random but rather a sequence of numbers that simulate randomness based on a "seed" number. Typically this formulas use modulos.

Once you've your uniform random draw you just need to use .

In step b), generate similarly a random draw from the distribution and accept if outcome is bigger than a third uniform Outcome
Step c), just do as told.

To prove this, looks like a consequence of bayes theorem, i.e., p(x|y) = p(x) * p(y|x) / p(y), the details I leave them to you.
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