
Probability and Statistics Basic Probability and Statistics Math Forum 
 LinkBack  Thread Tools  Display Modes 
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: 1,977 Thanks: 1026 
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 $f(3) = 1,~P[3]=1/4$ $f(6)=f(7) = 2,~P[6]+P[7]=1/2+1/4 = 3/4$ 
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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Interpretation of Conditional Distribution of a Function of a Random Variable  John Travolski  Advanced Statistics  1  October 26th, 2017 08:42 PM 
Expected value of a function of a discrete random variable  snoopmt1  Probability and Statistics  1  August 24th, 2017 12:12 AM 
distribution of a function of a random variable problem  frankpupu  Advanced Statistics  2  March 1st, 2012 03:45 AM 
Discrete random variable  hoyy1kolko  Algebra  1  February 13th, 2011 05:32 AM 
Discrete random variable.  adbroadband  Algebra  1  January 31st, 2008 09:49 AM 