SOLUTION: The probability density function of a random variable X is f(x) = a.e^(-ax) if x > 0 and f(x) = 0 for all other values. Find the pth moment of the pdf?

Question 1151373: The probability density function of a random variable X is f(x) = a.e^(-ax)
if x > 0 and f(x) = 0 for all other values. Find the pth moment of the pdf?
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
The pth moment of a random variable X is defined as u*p = E(X^p).
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pdf is f(x) = a / e^(ax)
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Use the Moment generating theorem which tells us that the pth derivative of m(t) evaluated at t = 0 is the pth moment u*p of X.
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Note f(x) is an exponential distribution with parameter a.
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Now the moment generating function for t < a given f(x) is
:
m(t) = a/(a - t)
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This is great since m(t) doesn’t have to be defined for all t. We only need it to be defined for t near 0 because we’re only interested in its derivatives evaluated at 0.
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The pth moment of m(t) can be written as p!/a^p
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