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

<br /> a(X + B)^2 + C<br />

(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

<br /> \begin{align*}<br /> X + B &amp; \sim \text{n}(B,\sigma^2)\\<br /> (X+B)^2 &amp; \sim \chi^2(\delta)\\<br /> \intertext{(non-central chi-square)}<br /> \end{align*}<br />

In the end the expression the distribution

<br /> a(X+B)^2 + C<br />

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.