The population distribution of income in a populous developing country is assumed to be given (approximately) by the continuous PDF \(f(x ; \Theta)=\Theta(1+x)^{-(\Theta+1)} I_{(0, \infty)}(x)\) where \(\Theta>0\). A summary measure of a random sample outcome of size 100 from the population distribution is given by \(\sum_{i=1}^{100} \ln \left(1+x_{i}ight)=40.54\), where \(x_{i}\) is measured in 1,000 's of units of the developing countries currency.

(a) Define a pivotal quantity for the parameter \(\Theta\) (Hint: Theorem 10.13 and Problem 10.21 might be useful here).

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

(c) Calculate an outcome of a 90 level confidence interval for \(\Theta\) based on the expression derived in (b). Note: For large degrees of freedom \(>30\),

\(\chi_{V ; \alpha}^{2} \approx V\left[1-\frac{2}{9 V}+z_{\alpha}\left(\frac{2}{9 V}ight)^{1 / 2}ight]^{3}\), where \(\int_{z_{\alpha}}^{\infty} N(x ; 0,1) d x=\alpha\), is a very good approximation to critical values of the \(\chi^{2}\) distribution (M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972, p. 941)l.