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November 30th, 2017, 01:21 AM   #1
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Distribution of a Monotonic Function of a Discrete Random Variable

Suppose I have a discrete random variable $Y$ with PMF $f(y)$ and support $\lbrace 1, 2, ..., N \rbrace$. Suppose I define another discrete random variable $H=Floor(Log_2(Y))$. Floor is simply the function which returns the integer part of a value, so all decimal points are truncated.

Notice how this would be a simple application of the CDF technique if weren't for this floor function. After all, $Log_2(Y)$ is monotonically increasing. While it's also true that $Floor(Log_2(Y))$ is monotonically increasing on the support of $Y$, the inverse function isn't one to one because there are multiple values of $Y$ which result in the same value of $H$. However, I'm pretty sure there are ways to account for this, but I'm at a loss and have no idea where to go from here.

So, to sum things up, how would you derive the distribution of $H$?
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November 30th, 2017, 10:08 AM   #2
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with discrete distributions stuff like this is just bookkeeping.

$H$ has $M$ discrete values produced by $f(Y)$

$M\leq N$, they aren't the same because some values of $Y$ produce the same value of $H$

So what you have to do is take the values of $Y$ and form all the values of $H$

Then group the values of $H$ that are equal and sum up the probabilities of the $y$'s that produce them.

A simple example

$Y= \{(3,1/4),~(6, 1/2),~(7,1/4)\}$

$H=\{(1,1/4),~(2,3/4)\}$ as

$f(3) = 1,~P[3]=1/4$

$f(6)=f(7) = 2,~P[6]+P[7]=1/2+1/4 = 3/4$
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December 9th, 2017, 12:08 AM   #3
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Sorry for the late response on this one. Thank you very much, that was quite helpful despite not being quite as formal as I had initially hoped for. I found a fairly simple way to describe the distribution.
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