Question

A Pareto distribution is often used in economics to explain a distribution of wealth. Let a random variable X have a Pareto distribution with parameter θ so that its probability distribution function is

fx(2:0) = A.c-4

for and 0 otherwise. The parameters $$\theta>1$$ and are known and fixed; is a constant to be determined.

a) Assuming that $$\theta>3,$$ find the expected value and variance of ?

b) Show that for 3 ≥ θ > 2 the Pareto distribution has a finite mean but infinite variance, and for 2 ≥ θ > 1 both mean and variance are infinite.

Answer 1

(a) E(X) = A xrx-0 dx - 4 xr-O+1 % (r-o +1) Ixo E(X) = A x-0 0 -o (9 ) - Axo 0 (0-1) xo = (0-1) E (x2)= A x (0-1), co -(0-1) xo 260 (0-) - A (0-120 =0 (0-0) 0-1 X0 V(x) = E (+2) - 6²(x) - (01) - (0-1) 2 Ho for 2 <023 Let 0 = 3 A 2010-1 E(+2) = A x 202-01 2-0+1= 0 for 0 0 E(X2) does not ex o. v(x) 00 040-2 <1 so E (x) exists as the denominator in x 70.

1<O<2 E(x) = x so v(x) - x 1. < 0-2 < 0 it 022 as E(X) - 0.

8 AXO dx = 1 x-0+1 10 -oti = 1 I Xo - A xo-(0-1) X (0-1) (0-1) & A (0-1) X