Question

Let X1,...,Xm and Y1,...,Yn be two random samples, both from normal distribution. They have common variance σ^2 , and different mean μX,μY, respectively. Find the distribution of (Sx)^2/(Sy)^2, where (Sx)^2,(Sy)^2 are sample variances.

Answer 1

To find the distribution of (Sx)^2/(Sy)^2, we can use the F-distribution. The F-statistic is defined as:

F = (Sx)^2 / σ^2 / (Sy)^2 / σ^2

Since the samples are from normal distributions, the F-statistic follows an F-distribution with degrees of freedom (m-1) and (n-1) for X and Y respectively. Therefore:

F ~ F(m-1, n-1)

Now we can substitute (Sx)^2/(Sy)^2 into the F-distribution to get the distribution of (Sx)^2/(Sy)^2:

(Sx)^2/(Sy)^2 ~ F(m-1, n-1) / (n-1)

Therefore, the distribution of (Sx)^2/(Sy)^2 is an F-distribution with (m-1) and (n-1) degrees of freedom for X and Y, respectively, divided by (n-1).