S4 Square of Normal

We can use the change of variables formula to show that the square of a standard normal random variable is a chi-square with one degree of freedom. Suppose Z is N(0,1) -- standard Normal -- and Y=Z^2. What is the distribution of Y? We cannot immediately apply the change of variables formula, because this function is not one-to-one. Both negative and positive values of X are mapped into the same value of Y. This means that there is no unique inverse function, which we need to make the calculations for the change of variables. Both the positive and the negative square root of Y act as inverse functions when X is restricted to positive or negative numbers.

To solve this problem, we first make a change of variables to X=|Z|. X is the absolute value of Z. The density of X can easily be calculated because Z is symmetric. X combines negative and positive values of Z into the same positive value, and it has density which exactly twice the density of Z on the positive axis, and 0 on the negative axis. X takes only positive values, and now the transformation Y=X^2=Z^2 is one to one. So we can now apply the change of variables formula.