With the probability density function:
i am asked to find the k-th moment of X, i.e. .
How do i do this?
You can do this by parts, but there's a nice trick to doing this in general for all gamma densities.
Let
Then
And
Since that is a valid density in the integrand.
NOW you should know how the gamma function reduces (via parts)
and note that this is true even if k is not an integer.
Finally, let and and you have your answer.
I don't understand your question. In the end, you won't have anymore
baldeagle calculated the k-th moment of a gamma distribution with parameters and
Then just notice that if you substitute in the pdf of a gamma distribution, you'll get the pdf you have.
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Another method... :
Let's calculate the moment generating function of your distribution.
(defined for )
Then we know that the k-th moment of X will be the k-th derivative of its mgf, taken at t=0.
Find the Taylor series of when t is near 0 and its coefficients will be the derivatives.
which is like a geometric series... :
But the Taylor series of is
where denotes the k-th derivative of M taken at t=0.
By identification, it follows that