1. The Pareto distribution, with parameters ? and ?, has...

1. The Pareto distribution, with parameters ? and ?, has pdf f(x) = ??^? / x^(?+1), ? < x < ?, ? > 0, ? > 0. (a) Verify that f(x) is a pdf. (b) Derive the mean and variance of this distribution. (c) Prove that the variance does not exist if ? ? 2.

Transcript

00:01 Hi, i'm david and i'm here to have you on seeing your question.

00:03 In the question here, we are given the barital distribution with the density of the form, fx equal to the beta, alpha power beta, divided by the x power of the beta plus 1.

00:17 For the x inside the interval, from alpha to the infinity, and alpha beta greater than 0.

00:25 In the first question, i will need to verify this one will be the valid bdf.

00:29 It means then we need to do integral from alpha to infinity, the density beta, alpha, beta over x power beta plus 1, the x.

00:41 Then we have the beta, alpha, beta will be the constant i can bring outside, and far to infinity.

00:48 I can bring that on the top, we have the minus beta minus 1, and then the x, and then untidy derivative of that we will have plus 1 to the power.

00:59 And we have x to the power minus beta, divided by the minus beta, evaluating from the alpha to the infinity.

01:08 Here we can cross out the beta with the beta, and then it will put infinity inside, we've got the 0, and it will put the alpha inside we have equal to the plus, and this one will be not the e, this one will be the alpha here.

01:25 So it will be the alpha, power beta times under the alpha.

01:30 Alpha by minus beta and equal to 1.

01:32 So therefore it is valid bdf.

01:39 And then for the question b, we need to find the mean and the variance...

Question

1. The Pareto distribution, with parameters ? and ?, has pdf f(x) = ??^? / x^(?+1), ? < x < ?, ? > 0, ? > 0. (a) Verify that f(x) is a pdf. (b) Derive the mean and variance of this distribution. (c) Prove that the variance does not exist if ? ? 2.

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