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Moment generating function for simple distribution
Posted:
Jan 12, 2011 7:01 PM


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



