Chapter 5: Q9E (page 306)

Consider the following probability distribution:
a. Calculate for this distribution.
b. Find the sampling distribution of the sample mean for a random sample of n = 3 measurements from this distribution, and show that is an unbiased estimator of .
c. Find the sampling distribution of the sample median for a random sample of n = 3 measurements from this distribution, and show that the median is a biased estimator of .
d. If you wanted to use a sample of three measurements from this population to estimate , which estimator would you use? Why?

Short Answer

Expert verified

$μ = 5$
is not an unbiased estimator of .
m is not an unbiased estimator of .
None

Step by step solution

Calculation of the mean μ in part (a)

The calculation of the mean $μ$ in case of the three values of x is shown below:

$μ = \underset{}{\sum x p (x)} = 2 (\frac{1}{3}) + 4 (\frac{1}{3}) + 9 (\frac{1}{3}) = \frac{2}{3} + \frac{4}{3} + \frac{9}{3} = \frac{15}{3} = 5$

Therefore the value of $μ$ is 5.

Determining whether x is an unbiased estimator of μ

$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$

The list of medians along with the associated probabilities is shown below:

Samples	Medians	Probability
2,2,2	2	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
2,4,2	2	localid="1658206263651" $\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
2,9,2	2	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
2,2,4	2	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
2,2,9	2	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
2,4,4	4	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
2,9,9	9	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
4,4,4	4	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
4,2,4	4	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
4,9,4	4	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
4,4,2	4	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
4,4,9	4	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
4,2,2	2	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
4,9,9	9	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
9,9,9	9	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
9,2,9	9	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
9,4,9	9	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
9,9,2	9	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$ $\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
9,9,4	9	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
9,2,2	2	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
9,4,4	4	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
2,4,9	4	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
2,9,4	4	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
4,2,9	4	$\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$
4,9,2	4
9,2,4	4
9,4,2	4

The summation of the medians are shown below:

As is 4.78 and is 5, m is not an unbiased estimator of .

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Short Answer

Step by step solution

Calculation of the mean μ in part (a)

Determining whether x is an unbiased estimator of μ

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