# E[x^(-1/2)] of exponential distribution

• September 20th 2009, 07:08 AM
MJM
E[x^(-1/2)] of exponential distribution
have a hard time to calculate the expected value below
E[x^(-1/2)], x is an exponential distribution of beta. I tried to transformed to a gamma but not succeed. the interal is way too big for me figure out. Thanks.
• September 21st 2009, 05:20 AM
Laurent
Quote:

Originally Posted by MJM
have a hard time to calculate the expected value below
E[x^(-1/2)], x is an exponential distribution of beta. I tried to transformed to a gamma but not succeed. the interal is way too big for me figure out. Thanks.

You need to compute $\int_0^\infty \frac{\beta e^{-\beta x}}{\sqrt{x}}dx$. You should try to substitute $u=\sqrt{x}$; it will give you an integral involving the Gaussian density function. From there you can conclude.
• September 21st 2009, 05:56 AM
MJM
Thanks. It works nicely.