Question

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

1. $\mathrm{\mu }=5$
2. is not an unbiased estimator of .
3. m is not an unbiased estimator of .
4. None

Step-by-step solution

Calculation of the mean μ in part (a)

The calculation of the mean$\mathbf{\mu }$in case of the three values of x is shown below:

$$\begin{gathered} \mu =\underset{}{\sum xp\left(x\right)} \\ =2\left(\frac{1}{3}\right)+4\left(\frac{1}{3}\right)+9\left(\frac{1}{3}\right) \\ =\frac{2}{3}+\frac{4}{3}+\frac{9}{3} \\ =\frac{15}{3} \\ =5 \end{gathered}$$

Therefore the value of $\mathbf{\mu }$is 5.

Determining whether x is an unbiased estimator of μ 

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

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 .