- #1
Mogarrr
- 120
- 6
Homework Statement
I'm trying to show that the definite integral:
[itex] \int_0^{\infty} \frac 1{\sqrt{2 \pi}} \sqrt {y} e^{\frac {-y}2} dy [/itex],
equals 1.
Homework Equations
it's already known that [itex] \int_0^{\infty}\frac 1{\sqrt{2 \pi}} y^{\frac {-1}2} e^{\frac {-y}2} dy = 1 [/itex], since f(x) is a probability density function.
The Attempt at a Solution
I've tried integration by parts, but that hasn't helped.
[itex] \int_0^{\infty} \frac 1{\sqrt{2 \pi}} \sqrt {y} e^{\frac {-y}2} dy = \frac 1{\sqrt{2 \pi}} (-2 \sqrt {y} e^{\frac {-y}2})|_0^{\infty} + \int_0^{\infty}\frac 1{\sqrt{2 \pi}} y^{\frac {-1}2} e^{\frac {-y}2} dy [/itex],
but [itex]lim_{t \to \infty} -2 \sqrt {y} e^{\frac {-y}2})|_0^{\infty} = 0 [/itex], so that's not very helpful.
Any ideas to evaluate the integral?