iid normals

Another quick post today because I really, really, really, need to finish off my CISS paper soon.

Let X₁, ..., X_n be a random sample from a N(μ, σ²) distribution. Then

• The sample mean and variance are independent random variables.
• The sample mean is distributed N(μ, σ²/n).
• (n - 1)S² / σ² is distributed chi-squared with n - 1 degrees of freedom.

That last one is exactly the type of thing that would show up on the Q.