My Math Forum  

Go Back   My Math Forum > High School Math Forum > Probability and Statistics

Probability and Statistics Basic Probability and Statistics Math Forum

Thanks Tree1Thanks
  • 1 Post By romsek
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 $H$?
John Travolski is offline  
November 30th, 2017, 09:08 AM   #2
Senior Member
romsek's Avatar
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

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

$f(6)=f(7) = 2,~P[6]+P[7]=1/2+1/4 = 3/4$
Thanks from John Travolski
romsek is online now  
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.
John Travolski is offline  

  My Math Forum > High School Math Forum > Probability and Statistics

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

Copyright © 2019 My Math Forum. All rights reserved.