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Topic: Moment generating function for simple distribution
Replies: 2   Last Post: Jan 14, 2011 2:58 PM

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Alexander Solla

Posts: 9
Registered: 12/17/10
Moment generating function for simple distribution
Posted: Jan 12, 2011 7:01 PM
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Hi everybody,

Can somebody check my work? I'm becoming more confident that I am
right and Schaum's notes on probability are wrong. ;0)

The question: Find a moment generating function for the random
variable defined by:

X = 1/2 with probability 1/2
X = -1/2 with probability 1/2

Compute the first four moments about the origin.

The work: We can compute the MGF as the sum of (exp(tx)P{x}) over all
x's "in" the random variable X. In this case, this comes to:

M_X(t) = 1/2 * (exp(t/2) + exp(-t/2))

Now we can compute moments about the origin by computing derivatives:
the kth moment is given by the kth derivative (with respect to t) of
the moment generating function, evaluated at t = 0. For example, the
first moment -- the mean -- is given by:

mu = d/dt (1/2 * (exp(t/2) + exp(-t/2)) at t = 0
= 1/4 * (exp(t/2) - exp(-t/2)) at t = 0
= 1/4 * ((exp(0) - exp(0))
= 0

This looks eminently reasonable, especially since the mean is
obviously 0.

Now I'm trying to compute the second moment about the origin:

mu_2 = d^2/dt^2 (1/2 * (exp(t/2) + exp(-t/2))) at t = 0
= d/dt 1/4 * (exp(t/2) - exp(-t/2)) at t = 0
= 1/4 * (1/2 exp(t/2) + 1/2 exp(-t/2)) at t = 0
= 1/8 * (exp(t/2) + exp(-t/2)) at t = 0
= 1/8 * (exp(0) + exp(0)) = 1/8 * 2 = 1/4

The answer key I'm using says this ought to be 1. Am I missing
something?

Thanks,
Alex



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