23. The population distribution of income in a populous developing country is assumed to be given (approximately)

by the continuous PDF f(x; Y) ¼ Y (1+x)

(Y+1) I(0,1)(x)

where Y > 0. A summary measure of a random sample outcome of size 100 from the population distribution is given by P100 i¼1 ln 1ð Þ¼ þ xi 40.54, where xi is measured in 1,000’s of units of the developing countries currency.

(a) Define a pivotal quantity for the parameter Y (Hint:

Theorem 10.13 and Problem 10.21 might be useful here).

(b) Define a general expression for a level g confidence interval for Y based on n iid observations from f(x; Y).

(c) Calculate an outcome of a .90 level confidence interval for Y based on the expression derived in (b). Note:

For large degrees of freedom > 30, w2 v;a v 1  2 9v þ za 2

9v

1=2 " #3

;

where Ð 1 za N(x; 0,1) dx ¼

a, is a very good approximation to critical values of the w2 distribution (M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972, p. 941)).