Statistics 6401, Fall 2012 1

Extra Credit

Direct proof that X̄ and S^{2} are independent when sampling from the N(μ, σ^{2}) distribution. Let X_{1}, X_{2} independent N(μ, σ^{2}) random variables. (A random sample of size n = 2.)

1. Show that Y_{1} = X_{1} + X_{2} and Y_{2} = X_{2} − X_{1} are independent.

2. What is the distribution of Y_{1}? What is the distribution of Y_{2}?

3. Show that W_{1} = \frac{1}{2}Y_{1} and W_{2} = \frac{1}{2}Y_{2} are independent.

Note:

W_{1} = \frac{1}{2}Y_{1}
= \frac{1}{2}(X_{1} + X_{2})
= X̄

and

W_{2} = \frac{1}{2}Y_{2}
= \frac{1}{2}(X_{2} − X_{1})
= \frac{1}{2}X_{2} − \frac{1}{2}X_{1}
= X_{2} − \frac{1}{2}X_{1} − \frac{1}{2}X_{2}
= X_{2} − (\frac{X_{1} + X_{2}}{2} )
= X_{2} − X̄

(So X̄ and the n − 1 deviations from the sample mean are independent.)

4. What is the distribution of W_{1}? What is the distribution of W_{2}?

5. Show that since W_{1} and W_{2} are indenpendent that W_{3} = X_{1} − X̄ is also independent of W_{1}. Note:

X_{1} − X̄ + X_{2} − X̄ = 0

(So X̄ and the first deviation X_{1} − X̄ are also independent.)

6. Argue that X̄ and S^{2} = \frac{1}{n−1} ∑_{i=1}^{2}(X_{i} − X̄)^{2} are independent for a random sample of size n = 2 from the N(μ, σ^{2}) distribution.

7. What is the distribution of X̄? What is the distribution of S^{2}?

8. Develop the same results for n = 3.

9. Develop the same result for a sample of size n.