Proofs related to chi-squared distribution for k degrees of freedom

Question

I was reading a proof related to chi-squared distribution for k degrees of freedom from wiki.

https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution

I think I might understand the general idea behind the proof. But there are some subtle details which I am confused about.

1) What is the meaning of the notation $P(Q)dQ$? Shouldn't it just be $P(Q)$?

2)The integral $\int_vdx_1dx_2...dx_k$ is equal to the surface area of the (k − 1)-sphere times the infinitesimal thickness of the sphere which is $dR = dQ/2Q^{1/2}$. Why we need to times $dR$?

Could someone please help me with the questions?

Answer 1

I think the $dQ$ simply signifies that it is the probability density at a given point on $Q$ so you would need to integrate across some range of Q to assess the size of some probability.

The height of a probability distribution function represents the probability at any given infinitesimal point so to sum the total probability it is necessary to multiply the area by the height, hence times $dR$. If we use the discrete analogue of a dice, the height of the function at each point is $1/6$ so to sum the probability across any area we multiply the number of points by the height at each point which is 1/6.

$dR$ is like the height of $1/6$ and $6$ is like the total surface area (in this case 1-dimensional length).

Answer 2

1-) The meaning of ∫P(Q)*dQ is the probability of finding the value of Q, according to the limits of the integration.

2-) The use of dR in the proof is fundamental to simplify the description of the shield of the sphere. Since we are talking about a sphere, nothing more appropriate than use spherical coordinates. In cartesian coordinates, it would be very difficult. Then, we can replace the original variables xi's for R.

As I understood this proof. For a given value of xi's, such that Q=x1^2+x2^2+..+xN^2, the probability density is given by P(x1)P(x2)..P(xN), which is, for this case 1/(2*pi)^N*exp(-Q/2). Such probability density will be the same for all points in the shield of the sphere.

In statistics, the change of variables between y and x is always performed ∫P(y)*dy = ∫P(x)*dx with the correpondent limits of x and y.

So, any P(Q)*dQ should correspond to integrate P(xi's)*dxi's along the shield of the sphere.

Thus, ∫P(Q)*dQ = ∫∫..∫P(x1,x2,...,xN)*dx1.dx2...dxN, being the limits of the integral in xi's the correspondent to the shield of the N-sphere. However, it can be simplified, since P(x1,x2,...,xN)=1/(2*pi)^N*exp(-Q/2), and Q is constant and can go out of the integral. Thus:

∫P(Q)*dQ =1/(2*pi)^Nexp(-Q/2)∫∫..∫dx1.dx2...dxN

But ∫∫..∫dx1.dx2...dxN is nothing more than the volume of the shield, which can be admited as A*dR, where A is the area of N-sphere, and R is the radius, R=Q^0.5.

Then

P(Q)*dQ =1/(2*pi)^N*exp(-Q/2)AdR

The deduction follows as the link of Wikipedia.

https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution