The derivation of the formula for variance for a Pareto Distribution
I need help with the stepbystep process of how to derive the formula for variance.
First I need to see how to calculate E[X^2] and then how to get variance using Vax[X] = E[X^2]  (E[X])^2.
I'm pretty sure I'm doing something wrong with the integration of x^2*ab/(b+x)^a+1 and the algebra that follows.
I've been trying this for 2 days now and I cannot seem to get it.
I've posted a picture of my attempt so you can see what I'm expecting.
Answer
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Leave a comment if you need any clarifications.

How did you know to multiply and divide by(ak) lambda^(ak)????

We would like the function inside the integral to be a Pareto distribution, since we know it integrates to 1. So you try to create a Pareto distribution algebraic manipulations.

Shouldnt the end of be negative?? for when we integrate and then plug in infinity, that part goes to 0 as infinity is in the denominator but then it's minus a minus one?? To be more specific what is the step between a*lambda^a/(ak)/((ak)lambda^(ak) multiplied by the integration of (ak)lambda*^(ak)/x^(ak+1) dx and the last step. Because I get (a*lambda^a)/(ak)lambda^(ak) multiplied by 0  1

OOOO are you using the fact that the density integrates to 1 therefore in that last step for E[X^k] a is (ak)? then since it is the integral of the density it's just equal to 1!!

Yes, I revised my solution and added some more details.