E[X] and E[X^2] for function of Weibull RV

May 2010
2
0
I'm tryng to solve the following integrals:

1) (x/t)*(e^(-x/t))*(1+e^(-x/t))^-2

2) ((x^2)/t)*(e^(-x/t))*(1+e^(-x/t))^-2


Both on negative infinity to positive infinity.

You can see that they are both expectations of functions of a weibull random variable with theta of t and beta of 1.

1) Y = X(1+e^(-X/t))^-2
2) Y = (X^2)(1+e^(-X/t))^-2

I'm not sure if you need to use that or if I just can't solve the integrals properly. Any help?
 
Nov 2009
517
130
Big Red, NY
I'm tryng to solve the following integrals:

1) (x/t)*(e^(-x/t))*(1+e^(-x/t))^-2

2) ((x^2)/t)*(e^(-x/t))*(1+e^(-x/t))^-2


Both on negative infinity to positive infinity.

You can see that they are both expectations of functions of a weibull random variable with theta of t and beta of 1.

1) Y = X(1+e^(-X/t))^-2
2) Y = (X^2)(1+e^(-X/t))^-2

I'm not sure if you need to use that or if I just can't solve the integrals properly. Any help?
See here:

Wolfram Mathematica Online Integrator
 
Nov 2009
517
130
Big Red, NY
There may be some slick manipulation you can use.
You may not need it though.
Look at the definition of polylog.
Specifically, examine its behavior when you plug in \(\displaystyle \pm \infty.\)