Expectation of 1/Y for Uniform

Feb 2013
188
12
Toronto
Hi, I need to find the expection of 1/Y where Y is distributed by UNIF(1,2).

Can I do a jacobian transformation to find the pdf of 1/Y and then \(\displaystyle \int(1/Y)f_{1/Y}(Y)\) with limits 1 and 2?

Help much appreciated
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey MathJack.

You won't need a Jacobian transformation for this problem - your PDF will be 1 if in [1,2] and zero otherwise. Just use the normal formula for E[f(X)]
 
Feb 2013
188
12
Toronto
Thanks Chiro,

THough while I was studying for one of my subjects I saw that these transformations were used for finding the expectation of certain functions of Random variables. In particular proving unbiasedness, when finding UMVUE, can someone tell me the difference here, and why Jacobians are needed for such?

Cheers
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
The Jacobian is not, directly, a technique for random variables. It is a connected with a "change of variables" for integration. In this problem, as chiro said, the integral is \(\displaystyle \int \frac{1}{y} dy\). Also term "Jacobian" is primarily used for more than one variable, the one variable case not needing the complexity of the "Jacobian".
 
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