Most of the available text in estimation theory frequently harps on Gaussian probability density function which, otherwise an excellent mathematical device, often leads the reader not to explore properties of other pdfs. In Spring 2008, during my ECE 652 (Estimation and Filtering Theory) class, I came across this beautiful problem [1] which uses Gamma pdf in a classic estimation theory question. My solution for the problem goes like this ( code for the following courtesy CodeCogs):

[2],

[3].


References:
[1] Mendel J. M., “Lessons in estimation theory for signal processing, communications and control,” Prentice Hall, 1995, Problem 13-7.
[2] Integral of exponential functions.
[3] Gamma distribution.