document.write( "Question 1151373: The probability density function of a random variable X is f(x) = a.e^(-ax)
\n" ); document.write( "if x > 0 and f(x) = 0 for all other values. Find the pth moment of the pdf? \n" ); document.write( "

Algebra.Com's Answer #773131 by rothauserc(4718) $\"\"$ $\"About$

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The pth moment of a random variable X is defined as u*p = E(X^p).
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\n" ); document.write( "pdf is f(x) = a / e^(ax)
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\n" ); document.write( "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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\n" ); document.write( "Note f(x) is an exponential distribution with parameter a.
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\n" ); document.write( "Now the moment generating function for t < a given f(x) is
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\n" ); document.write( "m(t) = a/(a - t)
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\n" ); document.write( "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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\n" ); document.write( "The pth moment of m(t) can be written as p!/a^p
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