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Probability Density Function of a Quadratic Equation

  1. Sep 22, 2008 #1
    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
     
  2. jcsd
  3. Sep 28, 2008 #2

    statdad

    User Avatar
    Homework Helper

    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 [tex] X [/tex] as

    [tex]
    a(X + B)^2 + C
    [/tex]

    (note that [tex] B, C [/tex] are not the same numbers as the [tex] b, c [/tex] in your post.

    Since [tex] X \sim \text{n}(0,\sigma^2) [/tex] we can say

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

    In the end the expression the distribution

    [tex]
    a(X+B)^2 + C
    [/tex]

    can be described as a scaled (because of the multiplication by [tex] a [/tex]) and translated (due to the addition of [tex] C [/tex]) 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.
     
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