RevisionDojo

Question 1

The following frequency distribution table represents a data set:

Value ( $x$ )	4	5	6	7	9
Frequency ( $f$ )	8	12	6	8	2

Using this data set, calculate the unbiased estimator for the population variance.

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Question 2

Assume a distribution has parameters $a$ and $b$ satisfying $E(X) = a + b$ and $\mathrm{Var}(X) = a$ . Using the sample mean $\bar{x} = 5.6111$ and sample variance $s^2 = 1.8418$ from the data, find the method-of-moments estimators for $a$ and $b$ .

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Question 3

A statistical study calculated the sample variance of a set of data to be $s^2 = 1.8418$ .

Calculate the sample standard deviation $s$ . Give your answer correct to four decimal places.

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Question 4

Let $X_1, X_2, \dots, X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $a$ .

Consider the estimator for $b = \mu - a$ defined by $\hat{b} = \bar{X} - \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2$ .

It is given that $\hat{b}$ is an unbiased estimator for $b$ and that its variance is $\text{Var}(\hat{b}) = \frac{a}{n} + \frac{2a^2}{n-1}$ .

Show that the estimator $\hat{b}$ is consistent for $b$ .

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Question 5

Let $X_1, X_2, \dots, X_n$ be a random sample from a population with mean $a + b$ . Let $\bar{X}$ be the sample mean and let $\hat{a}$ be an unbiased estimator for $a$ .

Consider the estimator $\hat{b} = \bar{X} - \hat{a}$ . Determine whether $\hat{b}$ is an unbiased estimator for $b$ .

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Question 6

Let $X_1, X_2, \dots, X_n$ be a random sample from a distribution such that $E(X_i) = a + b$ and $\text{Var}(X_i) = a$ , where $a, b \in \mathbb{R}$ and $a > 0$ .

Determine whether the estimator $\tilde{b} = \bar{X} - \frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})^2$ is unbiased for $b$ . If it is biased, find its bias.

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Question 7

Let $X_1, X_2, \dots, X_n$ be a random sample of size $n$ from a normal distribution with mean $\mu$ and variance $a$ . Let $\bar{X}$ be the sample mean and $S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2$ be the unbiased sample variance. It is given that $\bar{X}$ and $S^2$ are independent.

Given $c = \frac{1}{n-1}$ , derive the variance of the unbiased estimator $\hat{b} = \bar{X} - c \sum_{i=1}^n (X_i - \bar{X})^2$ in terms of $a$ and $n$ .

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Question 8

Let $X_1, X_2, \dots, X_n$ be a random sample of size $n > 1$ from a distribution with mean $E(X_i) = a + b$ and variance $\text{Var}(X_i) = a$ , where $a$ and $b$ are constants such that $a > 0$ .

Find the constant $c$ such that the estimator $\hat{b}_c = \bar{X} - c \sum_{i=1}^n (X_i - \bar{X})^2$ is an unbiased estimator for $b$ .

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Question 9

Given the following data values with frequencies: 4 (8 times), 5 (12 times), 6 (6 times), 7 (8 times), and 9 (2 times), compute the unbiased estimator for the population mean.

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Question 10

Let $X_1, \dots, X_n$ be iid from a distribution with $\text{E}(X) = a + b$ and $\text{Var}(X) = a$ . Show that the estimator $\hat{a} = \frac{1}{n - 1} \sum_{i=1}^n (X_i - \bar{X})^2$ is unbiased for $a$ .

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Question 11

A sample of size $n = 36$ is taken from a population. For this data set, the sum of squared deviations from the mean is $\sum_{i=1}^{36} (x_i - \bar{x})^2 = 64.4640$ and the unbiased estimator of the population variance is $s^2 = 1.8418$ .

Compute the biased estimator of the variance (using denominator $n$ ) and find the numerical bias by comparing it with the unbiased estimator.

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Question Type 3: Estimating the mean and variance using unbiased estimators Exercises