i
= 1, ..., n Xi.Virtual Laboratories > Random Samples > 1 2 3 4 5 [6] 7 8 9
Suppose that (X1, X2, ..., Xn) is a random sample from the normal distribution with mean µ and standard deviation d. In this section, we will establish several special properties of the sample mean, the sample variance, and some other important statistics.
First recall that the sample mean is
M = (1 / n)i
= 1, ..., n Xi.
The distribution of the M follows easily from basic properties of independent normal variables:
1.
Show that M is normally distributed with mean µ and variance d2 / n.
2.
Show that Z = (M - µ) / (d / n1/2) has
the standard normal distribution.
The standard score Z will appear in several of the derivations below.
Recall that if µ is known, a natural estimator of the variance d2 is
W2 = (1 / n)i
= 1, ..., n (Xi - µ)2.
Although the assumption that µ is known is usually artificial, W2 is very easy to analyze and it will be used in some of the derivations below.
3.
Show that nW2 / d2 has the chi-square distribution with n degrees of
freedom.
4.
Use the result of the previous exercise to show that
Recall that the sample variance is
S2 = [1 / (n - 1)]i
= 1, ..., n (Xi - M)2.
The next series of exercises show that the sample mean M and the sample variance S2 are independent. First we will note a simple but interesting fact, that holds for a random sample from any distribution, not just the normal.
5.
Use basic properties of covariance to show that
for each i, M and Xi - M are uncorrelated:
Our analysis hinges on the sample mean M and the vector of deviations from the sample mean:
Y = (X1 - M, X2 - M, ..., Xn - 1 - M).
6.
Show that
Xn - M = -i
= 1, ..., n - 1 (Xi - M).
and hence show that S2 can be written as a function of Y.
7.
Show that the M and the vector Y have a joint multivariate normal distribution.
8.
Use the results of the previous exercises to show that M and the vector Y
are independent.
9.
Finally, show that M and S2 are independent.
We can now determine the distribution of the sample variance S2.
10.
Show that nW2 / d2 = (n - 1)S2
/ d2 + Z2 where W2 and Z
are as given above.
Hint: In the sum on the left, add and subtract M, and expand.
11.
Show that (n - 1) S2 / d2 has the
chi-squared distribution with n - 1 degrees of freedom.
Hint: Use the result of the previous exercise, independence, and moment
generating functions.
12.
Use the result of the previous exercise to show that
Of course, these are special cases of the general results obtained earlier.
The next sequence of exercises will derive the distribution of
T = (M - µ) / (S / n1/2).
13.
Show that T = Z / [V / (n - 1)]1/2. where Z
is as above and where V = (n - 1) S2 / d2.
14.
Use previous results to show that T has the t distribution with n - 1 degrees of
freedom.
The random variable T plays a critical role in constructing interval estimates for µ and performing hypothesis tests for µ.