Distribution of the Function of Random Variable (M.G.F. Technique)& Order Statistics

Transformations of Variables of the Continuous Type
In the preceding section we introduced the notion of a one-to-one transformation and the mapping of a set A onto a set B under that transformation. Those ideas were sufficient to enable us to find the distribution of a function of several random variables o f the discrete
type. In this lecture we shall examine the same problem when the random variables are of the continuous type, using the M.G.F. technique and according to the fact which states as follows"if two random variables X,Y have the same M.G.F. , then they have the same distribution, that is if
M_X (t)=M_Y (t),then F(x)=F(y)
It is again helpful to find the p.d.f. of statistic which can be defined as a function, for example 1)Y=?_(i=1)^n?X_i
2) Y=(?_(i=1)^n?X_i )/n
3) Y=(?_(i=1)^n??[X_i ?-?X ?]?^2)/(n-1)

a random samle and does not contain any unknown parameter.
Example
Let X_i~N(?,?^2 ),then find the distribution of
Y=(?_(i=1)^n?X_i )/n.
Solution