Editing Random variable (section)

===Example 3===

Suppose <math>X</math> is a random variable with a [[standard normal distribution]], whose density is

:<math> f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}.</math>

Consider the random variable <math> Y = X^2.</math> We can find the density using the above formula for a change of variables:

:<math>f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right|. </math>

In this case the change is not [[Monotonic function|monotonic]], because every value of <math>Y</math> has two corresponding values of <math>X</math> (one positive and negative).  However, because of symmetry, both halves will transform identically, i.e.,

:<math>f_Y(y) = 2f_X(g^{-1}(y)) \left| \frac{d g^{-1}(y)}{d y} \right|.</math>

The inverse transformation is
:<math>x = g^{-1}(y) = \sqrt{y}</math>
and its derivative is
:<math>\frac{d g^{-1}(y)}{d y} = \frac{1}{2\sqrt{y}} .</math>

Then,

:<math> f_Y(y) = 2\frac{1}{\sqrt{2\pi}}e^{-y/2} \frac{1}{2\sqrt{y}} = \frac{1}{\sqrt{2\pi y}}e^{-y/2}. </math>

This is a [[chi-squared distribution]] with one [[Degrees of freedom (statistics)|degree of freedom]].