Deriving of the chi-square density function

In summary, the conversation discusses the three steps to derive the chi-square density function. The first step involves considering a standard i.i.d. Gaussian random vector and its squared magnitude. The second step involves showing the recursive relation for the PDF of ||X_n+2||^2. The third step uses the recursive relation and the PDFs for n = 1 and n = 2 to find the density for n ≥ 3. The final result is the chi-square density function.
  • #1
jone
6
0
According to my textbook, to derive the chi-square density function, one should perform three steps. First we 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].

1. For n = 1, show that the PDF of [itex]||X_1||^2[/itex] is
[itex]f_{X^2_1}(x) = \frac{1}{\sqrt{2\pi x}}\exp\left(-\frac{x}{2}\right)[/itex]

I did step 1 using the CDF-method, so I don't need any help with that.

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]

I was thinking this 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.

3. Using the recursive relation as well as the PDF:s for n = 1 and n = 2, find the density for n ≥ 3.
 
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  • #2
I know that the final result is f_{X_n^2}(x) = \frac{1}{2^{n/2}\Gamma\left(\frac{n}{2}\right)}\exp\left(-\frac{x}{2}\right)x^{\frac{n}{2} - 1}but again, I'm not sure how to get there.
 

1. What is the chi-square density function?

The chi-square density function is a statistical distribution that is used to analyze and interpret the relationship between categorical variables. It is commonly used in hypothesis testing and goodness-of-fit tests.

2. How is the chi-square density function derived?

The chi-square density function is derived by squaring and summing the standard normal distributions of a set of independent variables. This results in a non-negative, right-skewed distribution with a single peak at the origin.

3. What are the assumptions of the chi-square density function?

The chi-square density function assumes that the variables being analyzed are categorical, the observations are independent, and the expected frequency of each category is at least 5.

4. How is the chi-square density function used in hypothesis testing?

The chi-square density function is used to determine the probability of obtaining a particular set of data if the null hypothesis is true. This probability, known as the p-value, is then compared to a predetermined significance level to determine if the null hypothesis can be rejected.

5. Can the chi-square density function be used for continuous data?

No, the chi-square density function is only applicable for categorical data. For continuous data, other statistical distributions such as the normal distribution or t-distribution should be used.

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