# Thread: Joint Probability Mean and Variance

1. ## Joint Probability Mean and Variance

I'm really confused about this problem, and I have to do it really really really soon (like tonight), so hopefully, someone is able to help me on this. Thank you so much!

So X1,...Xn are iid N(m,d^2) random variables, x-bar is sample mean and s^2 is sample variance.

So I already got the CI for m is (x-bar - t(n-1, alpha/2)s/(sqrt n), x-bar + t(n-1, alpha/2)s/(sqrt n)), and the CI for variance is ((n-1)s^2/(chi-squared (n-1, 1-alpha/2)), (n-1)s^2/(chi-squared, alpha/2).

I now have to find the probability that both m is an element of the CI of m and variance is an element of the CI of variance. So I know m and variance are not independent, so I think I have to find a joint probability density??? I'm really quite lost on this problem, and any help at all would be greatly appreciated. Thanks a lot!!!!

2. There is also another part to this question that I have been working on, but have been unsuccessful. So I have to find the confidence interval around variance with minimum length. I figured that I have to differentiate the length of the CI, namely the upper limit - the lower limit, and set that quantity to 0, but I am having a lot of trouble differentiating the 1/(chi-square) part....