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Old 04-20-2018, 03:52 PM
Handynasty Handynasty is offline
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Post Help with Gamma Random Variable Problem

Hi,

I run into this problem when I'm studying:

Suppose the continuous random variable X has the following pdf:

f(x) = (1/16)*(x^2)*e^(-x/2) if x > 0
0 otherwise

Find E(X^3).

Clearly the λ = 1/2 and α = 3, but I don't know how to solve it other than going through the complicated integration. Is there any good way to solve it?
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Old 04-20-2018, 03:58 PM
Abraham Weishaus Abraham Weishaus is offline
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Recognize the integrand as a gamma integrand with lambda=1/2 and alpha=6, figure out what the correct constant for that gamma is, and compare with the current constant (1/16).
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Old 04-20-2018, 04:25 PM
Handynasty Handynasty is offline
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I'm sorry Abraham, I don't quite follow.

I thought the lambda=1/2 and alpha=3. But even though, I still have to go through the tough integration by parts.
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Old 04-20-2018, 05:58 PM
Abraham Weishaus Abraham Weishaus is offline
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You want to calculate the third moment. That means integrating x^3 f(x). Notice that x^3, when multiplied by x^2, becomes x^5. So you're integrating x^5 e^{-x/2} times some constant. That looks like a gamma density function, doesn't it? And gamma density functions got to integrate to 1. So just figure out what the constant in the density function is and voila - the integral of x^5 e^{-x/2} got to be the reciprocal of that constant!
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Old 04-20-2018, 06:20 PM
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Michael Mastroianni Michael Mastroianni is offline
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Following Abrahamís method:



Hence the answer would be
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Old 04-24-2018, 01:13 PM
Handynasty Handynasty is offline
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Thank you all!
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