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 November 30th, 2017, 12:21 AM #1 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 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 $H$? November 30th, 2017, 09:08 AM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389 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 $f(3) = 1,~P=1/4$ $f(6)=f(7) = 2,~P+P=1/2+1/4 = 3/4$ Thanks from John Travolski December 8th, 2017, 11:08 PM #3 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 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. Tags discrete, distribution, function, monotonic, random, variable Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post John Travolski Advanced Statistics 1 October 26th, 2017 08:42 PM snoopmt1 Probability and Statistics 1 August 24th, 2017 12:12 AM frankpupu Advanced Statistics 2 March 1st, 2012 03:45 AM hoyy1kolko Algebra 1 February 13th, 2011 05:32 AM adbroadband Algebra 1 January 31st, 2008 09:49 AM

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