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Deriving the chi-square density function

  1. Jul 24, 2009 #1
    The problem statement, all variables and given/known data
    Consider a standard i.i.d. Gaussian random vector [itex]\mathbf{X} = [X_1 \cdot \cdot \cdot X_n]^T[/itex] and its squared magnitude
    [itex]||\mathbf{X}||^2 = \sum_{i = 1}^nX_i^2[/itex].
    According to my textbook, to derive the density function of a chi-square random variable [itex]||\mathbf{X}||^2 [/itex], one should perform three steps:

    1. For n = 1, show that the PDF is
    [itex]
    f_{X^2_1}(x) = \frac{1}{\sqrt{2\pi x}}\exp\left(-\frac{x}{2}\right)
    [/itex]

    2. For any n, show that the PDF satisfies the recursive relation
    [itex]f_{X_{n+2}^2}(x) = \frac{x}{n}f_{X_n^2}(x)[/itex]

    3. Using the recursive relation as well as the PDF:s for n = 1 and n = 2, find the density for n ≥ 3.

    The attempt at a solution

    I did step 1 using the CDF-method, so I don't need any help with that. I was thinking step 2 could be proven by induction and this is where I'm stuck. To prove by induction I first show the recursive relation for n = 1:
    [itex]f_{X_3^2}(x) = xf_{X_1^2}(x) = \squrt{\frac{x}{2\pi}}\exp\left(-\frac{x}{2}\right)[/itex].
    I tried to do that using the CDF method:
    [itex]
    P(X_1^2 + X_2^2 + X_3^2 < x) = \int_{x_1^2 + x_2^2 + x_3^2<x}\frac{1}{(2\pi)^{3/2}}\exp\left(-\frac{x_1^2 + x_2^2 + x_3^2}{2}\right)dx_1dx_2dx_3
    [/itex]

    Changing to spherical coordinates and integrating I get
    [itex]
    P(X_1^2 + X_2^2 + X_3^2 < x) = \frac{2}{\sqrt{2\pi}}\int_0^{\sqrt{x}}\exp\left(-\frac{r^2}{2}\right)r^2dr
    [/itex]

    Differentiating back gives
    [itex]
    f_{X_3^2}(x) = \frac{2x}{\sqrt{2\pi}}\exp\left(-\frac{x}{2}\right)
    [/itex]
    But this is not what the recursive relation gives, and I don't see why. Although I didn't obtain the correct result, next I have to show that
    [itex]f_{X_{m+1+2}^2}(x) = \frac{x}{m+1}f_{X_{m+1}^2}(x)[/itex].
    I'm not sure I know how to prove that. Maybe there is another way to show the recursive relation.
     
    Last edited: Jul 24, 2009
  2. jcsd
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