# Question

Use Theorem 8.11 to show that, for random samples of size n from a normal population with the variance σ2, the sampling distribution of S2 has the mean σ2 and the variance 2σ4/n–1. (A general formula for the variance of S2 for random samples from any population with finite second and fourth moments may be found in the book by H. Cramer listed among the references at the end of this chapter.)

Theorem 8.11

If and S2 are the mean and the variance of a random sample of size n from a normal population with the mean µ and the standard deviation s, then

1. and S2 are independent;

2. The random variable (n–1)S2/σ2 has a chi-square distribution with n – 1 degrees of freedom.

Theorem 8.11

If and S2 are the mean and the variance of a random sample of size n from a normal population with the mean µ and the standard deviation s, then

1. and S2 are independent;

2. The random variable (n–1)S2/σ2 has a chi-square distribution with n – 1 degrees of freedom.

## Answer to relevant Questions

Show that if X1, X2, . . . , Xn are independent random variables having the chi-square distribution with v = 1 and Yn = X1 + X2 + · · · + Xn, then the limiting distribution of As n → ∞ is the standard normal ...Find the percentage errors of the approximations of Exercises 8.26 and 8.28, given that the actual value of the probability (rounded to five decimals) is 0.04596. Verify that if T has a t distribution with v degrees of freedom, then X = T2 has an F distribution with v1 = 1 and v2 = v degrees of freedom. Find the mean and the variance of the sampling distribution of Y1 for random samples of size n from the population of Exercise 8.46. Use the result of Exercise 8.54 to find the sampling distribution of R for random samples of size n from the continuous uniform population of Exercise 8.46.Post your question

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