Hello everyone! I'm taking an Mathematical Stat course at university and I came up with this question.

Question

Question can be found via(the question involves hard to type notations, so I thought of uploading it and sharing: drive (dot) proton (dot) me / urls/FVHX8Y08RR#jb7eaDEeff3H

As for this question(the one I've marked in red colour), we need to find a parameter estimation for alpha.

I have completed part (i) and my immediate next approach for solving part (ii) was to find E(x) and E(x^2) and then find estimator for alpha.

However, here I'm facing trouble with finding E(x^2) using integration. I'm getting the answer for integration related to E(x^2) as follows: E(x^2) = [ alpha*((beta)^alpha)/(2-alpha) ]*[((infinity)^(2-alpha)) - ((beta)^(2-alpha))

Obviously alpha cannot be equal to 2 in this case given above.

Also, from part 1, we know E(x) will only hold when alpha>1.

Your help on finding a parameter estimator for alpha is highly appreciated!