Deriving the chi-square density function

[jone]

Homework Statement
Consider a standard i.i.d. Gaussian random vector $\mathbf{X} = [X_1 \cdot \cdot \cdot X_n]^T$ and its squared magnitude
$||\mathbf{X}||^2 = \sum_{i = 1}^nX_i^2$.
According to my textbook, to derive the density function of a chi-square random variable $||\mathbf{X}||^2$, one should perform three steps:

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

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

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:
$f_{X_3^2}(x) = xf_{X_1^2}(x) = \squrt{\frac{x}{2\pi}}\exp\left(-\frac{x}{2}\right)$.
I tried to do that using the CDF method:
$P(X_1^2 + X_2^2 + X_3^2 < x) = \int_{x_1^2 + x_2^2 + x_3^2<x}\frac{1}{(2\pi)^{3/2}}\exp\left(-\frac{x_1^2 + x_2^2 + x_3^2}{2}\right)dx_1dx_2dx_3$

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

Differentiating back gives
$f_{X_3^2}(x) = \frac{2x}{\sqrt{2\pi}}\exp\left(-\frac{x}{2}\right)$
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
$f_{X_{m+1+2}^2}(x) = \frac{x}{m+1}f_{X_{m+1}^2}(x)$.
I'm not sure I know how to prove that. Maybe there is another way to show the recursive relation.

 

[mighty2000]

Can you help me with this?

Hi there,

Thank you for sharing your attempt at solving this problem. It seems like you are on the right track, but there are a few things that could be improved to reach the desired result.

Firstly, in your attempt at step 2, you are using the CDF method to find the PDF. While this can work in some cases, it is not the most efficient way to prove the recursive relation. Instead, I would recommend using the definition of the PDF, which is the derivative of the CDF. This will allow you to directly manipulate the PDF and show the recursive relation.

Secondly, in your attempt at showing the recursive relation for n=1, there are a few errors. The first one is that the equation you wrote down is not the correct CDF for n=1. The correct CDF for n=1 is P(X_1^2<x) = \frac{1}{\sqrt{2\pi}}\int_0^{\sqrt{x}}\exp\left(-\frac{t^2}{2}\right)dt. Notice that I have replaced the x's in your equation with t's, as the integration variable should be different from the variable of the function you are integrating.

The second error is that you have used the wrong integration limits when changing to spherical coordinates. The correct limits are r=0 to \sqrt{x}, \theta=0 to \pi, and \phi=0 to 2\pi. I would recommend reviewing the process of changing to spherical coordinates to make sure you understand it fully.

Finally, to prove the recursive relation for n=m+1, you can use the result you have obtained for n=m. This will involve integrating over the variable x_m+1, and then using the result for n=m in the remaining integral. This should lead you to the desired result.

I hope this helps and good luck with your proof! Let me know if you have any further questions.