Two problems for MGF:
1) Let X be a random variable with moment generating function M(t) = [(2+e^t)/3]^9. Calculate the variance of X
2) Let the random variable X have moment generating function M(t) = e^(3t+t^2). What is E(X^2)?
Both of these require double derivatives requiring special rules I've long forgotten and the solution I have written out skips steps.
For 1), I actually got it right by noticing that it's a binomial variable with n = 9 and p = 1/3, so variance is np*(1-p) = 9 * 2 * 1 / 3 / 3 = 2, which is pure luck because I couldn't figure it out using the derivative. Same with #2
For 1), the first derivative is 9 / 3 * e^t * (2 + e^t) ^ 8 = 3 * e^t * (2 + e ^t) ^ 8, and from here I can't get the second derivative
For 2), I couldn't even get the first derivative, and I didn't recognize the MGF so couldn't use that trick
1) Let X be a random variable with moment generating function M(t) = [(2+e^t)/3]^9. Calculate the variance of X
2) Let the random variable X have moment generating function M(t) = e^(3t+t^2). What is E(X^2)?
Both of these require double derivatives requiring special rules I've long forgotten and the solution I have written out skips steps.
For 1), I actually got it right by noticing that it's a binomial variable with n = 9 and p = 1/3, so variance is np*(1-p) = 9 * 2 * 1 / 3 / 3 = 2, which is pure luck because I couldn't figure it out using the derivative. Same with #2
For 1), the first derivative is 9 / 3 * e^t * (2 + e^t) ^ 8 = 3 * e^t * (2 + e ^t) ^ 8, and from here I can't get the second derivative
For 2), I couldn't even get the first derivative, and I didn't recognize the MGF so couldn't use that trick
Refresher on multiple derivatives?
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