Math 445Mini-Assignment 2 Solutions
CH6.50
(WT; 10 points)
When sampling
from a normal distribution, we know 
has a Chi-squared distribution with (n 1) degrees of
freedom. For this Chi-squared distribution the mean is (n 1) and the variance
is 2(n 1).
- Then we know
. - We also know
. Since n is in the denominator of this variance,
the variance goes to 0 as n gets large. - By Equation (6.12), we know
. This reduces to
. And if n = 2, this further reduces to 
, by properties of the gamma function (see page
190 of the textbook).
Obviously,
. Hence, even though 
is an unbiased estimator of 
, S is not an unbiased estimator of 
.
CH6.64
(WT; 5 points)
Suppose 
are independent standard-normal random
variables.
a.
From a previous result in class, because the Zs are
independent standard normal random variables, we know 
has a chi-squared distribution with 4 degrees
of freedom. (Note: I could have chosen any 4 distinct Zs and the result would
still hold.)
b.
Let 
. Then T is a ratio of
a standard normal random variable and the square root of a chi-squared random
variable (see part a) divided by its
degrees of freedom. Furthermore, the numerator and denominator are independent
(note, 
was not included in the
chi-squared random variable created in part
a, so it is independent of that variable). Then by definition of a t random
variable, we know the quantity, T, has a t-distribution with 4 degrees of
freedom.
c.
Rename the variable created in part a as 
. Also, let 
. By the same result used in part a, 
has a chi-squared distribution with 6 degrees
of freedom. Furthermore, and are independent, because they were created
from independent standard-normal random variables. Then, by definition of an F
random variable, we know 
has an F distribution with 4 numerator
degrees of freedom and 6 denominator degrees of freedom.
e.
Recall our new textbook has a different parameterization of
the gamma distribution (compared to our textbook from last term).An exponential
random variable (mean=2) is also a gamma random variable with 
and 
. Also recall that a
chi-squared random variable (with v
degrees of freedom) is a special-case of a gamma random variable, where 
and . Hence, an exponential
random variable (mean=2) is also a chi-squared random variable with 2 degrees
of freedom. Then, by the same result used in part a, 
has a chi-squared
distribution with 2 degrees of freedom, and hence an exponential distribution
with mean 2. (Note: I could have chosen any 2 distinct Zs and the result would
still hold.)