Expectation of a density function

#1
Hi guys,

I have a question about the calculation of the expectation based on a density function where the integral is NOT 1.

Is it okay to normalize such a function and how would this work. Could anybody give a simple example?

Thanks!
 

Dason

Ambassador to the humans
#2
It doesn't really make sense to talk about expectation unless you have a probability measure. If the integral is not 1 you don't have a probability measure. What exactly are you doing?
 
#3
Yepp this was also my thought but I read the following: Let [tex] f(t) [/tex] be an function, no properties listed but I guess that [tex] \int \limits_0^\infty f(t) dt < \infty [/tex] then they define:

[tex] mean = \frac{\int \limits_0^\infty t f(t) dt}{\int \limits_0^\infty f(t) dt} [/tex]

[tex] var = \frac{\int \limits_0^\infty (t-mean)^2 f(t) dt}{\int \limits_0^\infty f(t) dt} [/tex]
 

Dason

Ambassador to the humans
#4
Tex tags are broken here but math tags work. And they normalize the function there so it should be fine (as long as f>0 for all x)