NW 5.10 Consider the following probability distribution: Xx p(x) 0 -102 3 1 com 3 2 mean and the sample median. f. Which estimator would you use to estimate ? Why? 3 a. Find μ. b. For a random sample of n = 3 observations from this distribution, find the sampling distribution of the sample mean. c. Find the sampling distribution of the median of a sample of n = 3 observations from this population. d. Refer to parts b and c, and show that both the mean and median are unbiased estimators of μ for this population. e. Find the variances of the sampling distributions of the sample

MATLAB: An Introduction with Applications
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only part c and e

## Probability Distribution Problem

### Consider the following probability distribution:

| \( x \) | 0   | 1   | 2   |
|---------|-----|-----|-----|
| \( p(x) \) | \( \frac{1}{3} \) | \( \frac{1}{3} \) | \( \frac{1}{3} \) |

---

### Questions:

a. **Find \( \mu \).**

b. **For a random sample of \( n = 3 \) observations from this distribution, find the sampling distribution of the sample mean.**

c. **Find the sampling distribution of the median of a sample of \( n = 3 \) observations from this population.**

d. **Refer to parts b and c, and show that both the mean and median are unbiased estimators of \( \mu \) for this population.**

e. **Find the variances of the sampling distributions of the sample mean and the sample median.**

f. **Which estimator would you use to estimate \( \mu \)? Why?**

---

This exercise involves working with a simple probability distribution and examining the properties of sample means and medians as estimators of the population mean \( \mu \).
Transcribed Image Text:## Probability Distribution Problem ### Consider the following probability distribution: | \( x \) | 0 | 1 | 2 | |---------|-----|-----|-----| | \( p(x) \) | \( \frac{1}{3} \) | \( \frac{1}{3} \) | \( \frac{1}{3} \) | --- ### Questions: a. **Find \( \mu \).** b. **For a random sample of \( n = 3 \) observations from this distribution, find the sampling distribution of the sample mean.** c. **Find the sampling distribution of the median of a sample of \( n = 3 \) observations from this population.** d. **Refer to parts b and c, and show that both the mean and median are unbiased estimators of \( \mu \) for this population.** e. **Find the variances of the sampling distributions of the sample mean and the sample median.** f. **Which estimator would you use to estimate \( \mu \)? Why?** --- This exercise involves working with a simple probability distribution and examining the properties of sample means and medians as estimators of the population mean \( \mu \).
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