Editing Random variable (section)

===Example 4===

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

:<math> f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\mu)^2/(2\sigma^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]], because every value of <math>Y</math> has two corresponding values of <math>X</math> (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:

:<math>f_Y(y) = f_X(g_1^{-1}(y))\left|\frac{d g_1^{-1}(y)}{d y} \right| +f_X(g_2^{-1}(y))\left| \frac{d g_2^{-1}(y)}{d y} \right|.</math>

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

Then,

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

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