Probability Density Function of a Quadratic Equation

[zudhirsharma]

HI Can anybody tell me how to calculate a PDF of y, where y is a function of x, such that
y = a X*X + bX + C (i.e. a quadratic equation), and X follows the Normal Distribution X ~N(0, sigma)

Help anybody?
Thanks

 

[statdad]

A rough (as in I haven't stepped through all of the work in the detail you'd need to turn in on an assignment).

First, complete the square to write your quadratic in $$X$$ as

$$a(X + B)^2 + C$$

(note that $$B, C$$ are not the same numbers as the $$b, c$$ in your post.

Since $$X \sim \text{n}(0,\sigma^2)$$ we can say

$$\begin{align*} X + B & \sim \text{n}(B,\sigma^2)\\ (X+B)^2 & \sim \chi^2(\delta)\\ \intertext{(non-central chi-square)} \end{align*}$$

In the end the expression the distribution

$$a(X+B)^2 + C$$

can be described as a scaled (because of the multiplication by $$a$$) and translated (due to the addition of $$C$$) noncentral chi-square. There is no name for this.

An alternate approach would be to attempt to calculate the characteristic function for your quadratic expression, then attempt inverting. I looked at that: it seemed less than exciting.