We now give some additional technical details about the chi-square distribution and provide proofs for some of the key propositions. Except for the proof of Corollary 2 knowledge of calculus is required.

**Observation**: For any positive real number *k*, per Definition 1, the chi-square distribution with *k* degrees of freedom, abbreviated χ^{2}(*k*), has probability density function

Using the notation of Gamma Function Advanced, the cumulative distribution function for *x *≥ 0 is

**Property A**: The moment generating function for a random variable with χ^{2}(*k*) distribution is

Proof:

Set y = *x*/2 (2θ–1). Then *x* = 2y/(1–2*θ*) and d*x*=2 dy/(1–2*θ*). Thus

The result follows from the fact that by Definition 1 of Gamma Function

**Property 1**: The χ^{2}(*k*) distribution has mean* k* and variance *2k*

Proof: By Property A, the moment generating function of χ^{2}(*k*) is

it follows that

**Theorem 1**: Suppose *x* has standard normal distribution *N*(0,1) and let *x _{1},…, x_{k} *be

*k*independent sample values of

*x*, then the random variable

has a chi-square distribution χ^{2}(*k*).

Proof: Since the *x _{i}* are independent, so are the . From Property 4 of General Properties of Distributions, it follows that

But since all the *x _{i}* are samples from the same population, it follows that

Since *x* is a standard normal random variable, we have

Set y = *x *. Then *x* = y/ and d*x* = dy/, and so

is the pdf for *N*(0, 1), it follows that

Combining the pieces, we have

But by Property A this is the same moment generating function as for χ^{2}(*k*). Thus, by Theorem 1 of General Properties of Distributions, it follows that *w* has distribution χ^{2}(*k*).

**Property 2**: If *x* and y are independent and *x* has distribution χ^{2}(*m*) and y has distribution χ^{2}(*n*), then *x* + y has distribution χ^{2}(*m + n*)

Proof: Since *x* and y are independent, by Property A and Property 4 of General Properties of Distributions

But this is the moment generating function for χ^{2}(*m + n*), and so the result follows from the fact that a distribution is completely determined by its moment generating function (Theorem 1 of General Properties of Distributions).

**Theorem 2**: If *x* is drawn from a normally distributed population *N*(*μ, σ*) then for samples of size *n* the sample variance *s*^{2} has distribution

Proof: Since (*x̄* – *µ*)^{2} is a constant and

Dividing both sides by *σ*^{2} and rearranging terms we get

Now we use the fact that *x* ~ *N*(*μ, σ*) with each *x _{i} *independently sampled from this distribution. Thus the left side of the equation is the sum of

*n*variables, each the square of a z-score, and so by Theorem 1, the left side of the equation has chi-square distribution with

*n*degrees of freedom:

Next consider the last term of the equation above. Since the sampling distribution of the mean for data sampled from a normal distribution *N*(*μ, σ*) is normal with mean *μ* and standard deviation *σ/*, it follows that the last term is the square of a z-score, and so by Theorem 1, it has chi-square distribution with 1 degree of freedom

By Property 2, it follows that the remaining term in the equation is also chi-square with *n* – 1 degrees of freedom.

Dividing by *n* – 1, it follows that

It follows that

which completes the proof.

**Corollary 2**: *s*^{2} is an unbiased, consistent estimator of the population variance

Proof (unbiased): From the theorem and Property 1 it follows that:

Proof (consistent): From Property 1 and Property 3b of Expectation:

This is refreshing and easy to follow for begginers. All too often many qualified/advanced staticians skip basic explanations and say ” …it is trivial to prove that… ” ; yet for the begginner, she/he would be expecting those very basics in order to come to a level where they will see the triviality! Thanks a lot and I will keep coming back for more as I am just starting statistics especially the mathematical statistics.

Sydeny,

Thanks for your comment. I try hard to give some of the theory and basic explanations, without getting too theoretical, for just the reasons you stated.

Charles

I loved your explanation, I’ve been looking in many books for this.

Thank you for this easy to follow proofs.

Thanks for your analysis I take it for my assignment

Excellent, helps me explain a lot of things!