Transformations

#1
Hi!

Suppose X ~ U(0,1). If we let Y = -ln (x) and then find the density of Y, we get the exponential distribution (with rate parameter = 1).

Okay,

I'm running into problems translating it into english.

If X is an RV on (0,1), and we pick some subset, say (0.25, 0.75) and take the -ln and of those values and plot them, I should have something resembling an exponential curve? Is there an exercise that I can do in R that might make this more apparent?

Any elaboration on this would be extremely helpful! Your insights are appreciated.
 
Last edited:

BGM

TS Contributor
#2
You can show that the truncated uniform is also uniformly distributed.
i.e. The uniform random variable truncated on the interval \( (0.25, 0.75) \)
\( X \sim Uniform(0.25, 0.75) \)
\( \Rightarrow f_X(x) = 2, x \in (0.25, 0.75) \)

Now we have \( Y = -\ln X \Rightarrow X = e^{-Y} \)
\( \Rightarrow f_Y(y) = f_X(e^{-y})\left|\frac {dx} {dy} \right|
= 2e^{-y}, y \in (-\ln 0.75, -\ln 0.25) \)

So it resemble another truncated exponential curve.
 

vinux

Dark Knight
#3
See the R code below:

Code:
x=runif(10000)
y=(x<.75) & (x>.25)

  par(mfrow=c(2, 1))
breaks <- pretty(range(x), 20)
hist(x[y==1], breaks=breaks,col='red', main="Hist of truncated Unif(RED)")
hist(x[y==0], breaks=breaks, add=TRUE, col='green')


lbreaks <- -1* log(breaks+.00001)
hist(-1*log(x[y==1]), breaks=lbreaks, col='red', main="Hist of truncated Exponential(RED)")
hist(-1*log(x[y==0]), breaks=lbreaks, add=TRUE, col='green')
*Note: I am not a good R programmer. Ecologist or Tart can help you for efficient way to write the above code
 
#4
Oh, fantastic. Thank you both!


I was also wondering if you knew of a source (book, course website, etc) that had something like an introduction / elementary probability with the use of a computer package for demonstration/simulation purposes.

I'm trying to fill in a lot of the gaps in my background before I start my masters this Sept.

Thanks again guys, you rock!

Cheers
 

mp83

TS Contributor
#5
I have used the book of Paolella "Intermediate Probability: A Computational Approach". It has Matlab & R code. Unfortunatelly, the price is high (~$160), but otherwise recommended... ;(

http://www.amazon.com/Intermediate-Probability-Computational-Marc-Paolella/dp/0470026375

The book:

  • Places great emphasis on the numeric computation of convolutions of random variables, via numeric integration, inversion theorems, fast Fourier transforms, saddlepoint approximations, and simulation.
  • Provides introductory material to required mathematical topics such as complex numbers, Laplace and Fourier transforms, matrix algebra, confluent hypergeometric functions, digamma functions, and Bessel functions.
  • Presents full derivation and numerous computational methods of the stable Paretian and the singly and doubly non-central distributions.
  • A whole chapter is dedicated to mean-variance mixtures, NIG, GIG, generalized hyperbolic and numerous related distributions.
  • A whole chapter is dedicated to nesting, generalizing, and asymmetric extensions of popular distributions, as have become popular in empirical finance and other applications.
  • Provides all essential programming code in Matlab and R.
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