linguistic:translation
Example 4

Suppose X {\displaystyle X} is a random variable with a normal distribution, whose density is

Consider the random variable Y = X 2. {\displaystyle Y=X^{2}.} We can find the density using the above formula for a change of variables:

In this case the change is not monotonic, because every value of Y {\displaystyle Y} has two corresponding values of X {\displaystyle X} (one positive and negative). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms:

The inverse transformation is

and its derivative is

Then,

This is a noncentral chi-squared distribution with one degree of freedom.