28. Let X have the Pareto pdf introduced in Exercise...

28. Let X have the Pareto pdf introduced in Exercise 10: f(x; k, ?) = { (k · ??) / x??¹ x ? ?; 0 x < ? (a) If k > 1, compute E(X). (b) What can you say about E(X) if k = 1? (c) If k > 2, show that Var(X) = k?²(k - 1)?²(k - 2)?¹. (d) If k = 2, what can you say about Var(X)? (e) What conditions on k are necessary to ensure that E(X?) is finite?

Transcript

00:01 Okay, so if k is greater than 1, compute the expectation of x.

00:05 So the expectation of x is just x times the pdf integrated over all allowed values, so that's from theta up to infinity.

00:14 So if we plug in our expression for f, we find that this is k times theta to the k over x to the power k.

00:25 Now for k greater than 1, which it is, we're told it is in this case, then this just integrates to minus 1 over k plus 1 times theta to the k over x to the k plus 1.

00:46 Sorry, this is k minus 1, not k plus 1.

00:48 We're integrating, so the power of x is getting more closer to 0, because it's minus k.

00:58 This is x to the minus k, so integrate it, you get minus x to the minus k minus 1.

01:05 And then we evaluate that between theta and infinity, that's a theta not a 0.

01:11 And at infinity, this term is obviously just going to go to 0, so we just get 0 minus minus k over k minus 1 times theta to the k over theta to the k minus 1, which is just theta.

01:26 So we get k over k minus 1 theta.

01:30 So that's the expectation of x.

01:32 And part b says, what can you say about the expectation of x if k equals 1? well, if k equals 1, then we get the integral from theta to infinity of k theta to the k, sorry, of just theta over x, which is theta ln x evaluated between infinity and theta.

01:55 And clearly that is undefined, it's an infinite answer.

02:03 Part c says, find the variance of x if k is bigger than 2.

02:08 So remember, the variance is the expectation of x squared minus the expectation of x squared...

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