The integral
Can be rewritten as
If you replace
So both integrals are the same this gives
let
and we have that
This can be evaluated by integration by parts, but it is also the the gamma function evaluated at n+1
So this gives
Hi Guys,
I need help with a problem I cannot get past. The question below gives me the hint to make the substitution to make the integration easier and I understand that I the area under the curve must be one which will help me solve for c. But I can't see how the original equation simplifies to the integral the question provides. Could someone please show me how the original equation simplifies to the given integral once the substitution has been made. Once I understand that I can solve for c.
Thanks!
Nappy
The integral
Can be rewritten as
If you replace
So both integrals are the same this gives
let
and we have that
This can be evaluated by integration by parts, but it is also the the gamma function evaluated at n+1
So this gives
How would I go about calculating the first moment of the above equation? I tried both ways by first integrating with the multiplication of x to the integral and the other way by tring to calculate the moment generating function and setting t=0 for the first moment. I keep getting to a point where my answer is (n+1/2)! Is it possible to have a (n+1/2)! factorial. The calculator reads error. I'd really appreciate if someone could show me how to calculate the first moment. I need to find all moments. Thanks for the help!
To find the kth moment we multiply the integrand by
Using the integral from post #2
Using the same substitution we get
This gives
This integral is the gamma function. This generalizes the factorial function.
Gamma function - Wikipedia, the free encyclopedia
This gives
This values can be calculated for any value of n and k