SOLUTION: I have a homework question that has me stumped. Desperatedly need help. A random variable X has moment generating function M(t) = (1-t)^-1. Find the rth moment of X.

Algebra ->  Probability-and-statistics -> SOLUTION: I have a homework question that has me stumped. Desperatedly need help. A random variable X has moment generating function M(t) = (1-t)^-1. Find the rth moment of X.      Log On


   

Question 349587: I have a homework question that has me stumped. Desperatedly need help.
A random variable X has moment generating function M(t) = (1-t)^-1. Find the rth moment of X.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
M%28t%29=1%2F%281-t%29
Look at the derivatives of M%28t%29 evaluated when t=0.
The rth moment is the rth derivative of M%28t%29 evaluated at t=0.
1st derivative:M1%28t%29=%281-t%29%5E%28-2%29
M1%280%29=1
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2nd derivative:M2%28t%29=2%281-t%29%5E%28-3%29
M2%280%29=2
3rd derivative:M3%28t%29=6%281-t%29%5E%28-4%29
M3%280%29=6
4th derivative:M4%28t%29=24%281-t%29%5E%28-5%29
M4%280%29=24
5th derivative:M5%28t%29=120%281-t%29%5E%28-6%29
M5%280%29=120
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Hopefully you see the pattern here,
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1st derivative: 1=1%21
2nd derivative: 1%2A2=2%21
3rd derivative: 1%2A2%2A3=3%21
4th derivative: 1%2A2%2A3%2A4=4%21
5th derivative: 1%2A2%2A3%2A4%2A5=5%21
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rth derivative: 1%2A2%2A3%2A4%2A5*...*%28r-1%29%2Ar=r%21
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So the rth moment of M%28t%29 is r%21.