# Question: Prove Theorem 8 10 Theorem 8 10 If X1 and X2 are independent

Prove Theorem 8.10.

Theorem 8.10

If X1 and X2 are independent random variables, X1 has a chi-square distribution with v1 degrees of freedom, and X1 + X2 has a chi-square distribution with v > v1 degrees of freedom, then X2 has a chi-square distribution with v – v1 degrees of freedom.

Theorem 8.10

If X1 and X2 are independent random variables, X1 has a chi-square distribution with v1 degrees of freedom, and X1 + X2 has a chi-square distribution with v > v1 degrees of freedom, then X2 has a chi-square distribution with v – v1 degrees of freedom.

## Relevant Questions

Verify the identity Which we used in the proof of Theorem 8.11. If the range of X is the set of all positive real numbers, show that for k > 0 the probability that √2X – √2v will take on a value less than k equals the probability that X – v / √2v will take on a value less than ...Verify that if X has an F distribution with v1 and v2 degrees of freedom and v2 → ∞, the distribution of Y = v1X approaches the chi-square distribution with .1 degrees of freedom. Find the sampling distributions of Y1 and Yn for random samples of size n from a continuous uniform population with α = 0 and β = 1. Use the formula for the joint density of Y1 and Yn shown in Exercise 8.52 and the transformation technique of Section 7.4 to find an expression for the joint density of Y1 and the sample range R = Yn – Y1.Post your question