A family of pdf\'s that has been used to approximate the distribution of income,

Question

A family of pdf's that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, k and theta¸ both > 0, and the pdf is the following. f(x: k, theta) = {k middot theta^k/x^k + 1 x greaterthanorequalto theta 0 x 1, compute E(X). E(X) = ____ (b) What can you say about E(X) if k = 1? E(x) is equal to 0. E(X) is equal to 1. E(X) is undefined. E(X) cannot be described without first defining theta. (c) If k > 2, show that V(x) = k theta ^2(k - 1)^-2(k-2)^-1. E(X^2) = k theta^k integral_ theta ^infinity ______ dx = ___ Therefore, V(X) = (_______/k - 2) - (k theta)/_____)^2 = k theta ^2/(k - 1)^2 (k - 2) (d) If k = 2, what can you say about V(X)? V(x) is equal to 0. V(X) is equal to 1 V(X) is undefined. V(X) cannot be described without first defining theta. (e) What conditions on k are necessary to ensure that E(X^n) is finite? E(X^n) is finite if and only if n-k. E(X^n) is finite if and only if n k.

Explanation / Answer

Solution

Given, f(x, k, ) = kk(x- k – 1), x

                          = 0, x <

Back-up Theory

E(X) = integral[, ]{x.f(x)} ……………………………………………… (1)

E(X2) = integral[, ]{x2.f(x)} ……………………………………………… (2)

V(X) = E(X2) – {E(X)}2 …………………………………………………….(3)

Part (a)

[vide (1) of Back-up Theory], E(X)

= integral[, ]{x. kk(x- k – 1)dx}

= kk. integral[, ](x- kdx)

= kk{x- k + 1/(1 – k)}[, ]

= kk{- - k + 1/(1 - k)}

= kk{ - k + 1/(k – 1)}

= k/(k – 1) ANSWER

Part (b)

At k = 1, (k – 1) = 0 and hence E(X) is undefined. Option (3) ANSWER

Part (c)

[vide (2) of Back-up Theory], E(X2)

= integral[, ]{x2. kk(x- k – 1)dx}

= kk.integral[, ](x- k + 1dx)

= kk{x- k + 2/(2 – k)} , ]

= kk{- - k + 2/(2 – k)}

= k2/(k – 2)

So, [vide (3) of Back-up Theory], V(X)

= {k2/(k – 2)} – {k/(k – 1)}2

= k2[{1/(k - 2)} - {k/(k – 1)2}

= k2[(k – 1)2 - k(k – 2)]/{(k - 2)(k – 1)2}

= (k2)/{ (k - 2)(k – 1)2}

= (k2)(k - 2)- 1(k – 1)- 2} PROVED

Part (d)

If k = 2, (k - 2) = 0 and hence V(X) is undefined. Option (3) ANSWER

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