Two efficiency experts take independent measurements Y1 and Y2 on the length of time workers take to complete a certain task. Each measurement is assumed to have the density function given by
f(y) = { (1/4)y*e^(-y/2) for y > 0, 0 elsewhere
Find the density function for the average U = (1/2)(Y1 + Y2).
How can we solve this problem by using the method of moment-generating functions?
Moment generating function is defined as $M_X(t) = E[e^{tX}]$, where $t$ is a real number. This is a common approach to find the distribution of sum of independent random variables as it uniquely determines that. Any standard textbook on Mathematical Statistics would have fair details on it. Once you know the MGF form of any random variable you can determine the corresponding distribution (for this you need to be bit familiar with MGF form that corresponds to a distribution). Try finding the MGF of Y first and then see how the MGF of U comes out to be. [NB: The given pdf for $Y$ above is a chi-square with $4$ d.f.]