(a) Let X be a continuous random variable with probability...

(a) Let X be a continuous random variable with probability density function f_X and moment generating function ?_X(t) := E(e^{tX}) defined for all t ? R. Let Y = e^X. Prove that the n-th moment of Y is equal to ?_X(n), where n ? N. (b) Let U be a random variable distributed according to the log-normal model with parameters ? and ?^2 (i.e., log(U) ~ N(?, ?^2)). Obtain the variance of U. (Hint: If X ~ N(?, ?^2), then ?_X(t) = e^{?t + ?^2t^2/2}).

Transcript

00:01 So in this question we have x, which is a continuous random variable with a probability density function f of x and a moment generating function phi x of t is the expected value of e to the t x for any t in r.

00:25 Now if we have y is e to the x, then what we want to do is we want to prove that the nth moment of y is equal to phi x of n for n a natural number.

00:46 Now the nth moment of y is the expected value of y to the power of n, which is the expected value of e to the x to the power of n.

00:58 But that's the expected value of e to the n x, but let's just compare that with phi x of t.

01:05 That's the expected value of e to the t x, so this is straightforwardly phi of n.

01:15 So it really is a one -line proof and we're done.

01:19 For part b, u is a random variable with a log -normal distribution.

01:26 So log u is normal with a mean of mu and a variance of sigma squared.

01:35 We want the variance of u, so that's the expected value of u squared minus the expected value of u squared.

01:45 But if we say that x is e to the power of u, sorry, e to the power of u, actually no, we want to say x is log u, so that means that x is normal, but it also means that u is e to the x.

02:08 Now from above, we know that when we have something which is e to the x, its nth moment is phi x of n.

02:16 So that means the expected value of u to the n is phi x of n.

02:22 But we also know for x normal mu sigma squared, phi x of t is e to the mu t plus sigma squared t squared over 2.

02:35 So that means the expected value of u to the n is e to the mu n plus sigma squared n squared over 2.

02:50 So now we can get the variance...

Question

Please give Ace some feedback