Problem 3 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

\(\square\) Suppose the random variables \(X_{1}, X_{2}, \ldots, X_{n}\) have the same expectation \(\mu\). For which constants \(a\) and \(b\) is $$ T=a\left(X_{1}+X_{2}+\cdots+X_{n}\right)+b $$ an unbiased estimator for \(\mu ?\)

Step-by-Step Solution

Answer

a = \( \frac{1}{n} \), b = 0.

Step 1: Define Unbiased Estimator

An estimator \( T \) is said to be unbiased for the parameter \( \mu \) if the expectation of \( T \) is equal to \( \mu \). Therefore, the condition for \( T \) to be an unbiased estimator is \( E[T] = \mu \).

Step 2: Express the Expectation of T

The expression for \( T \) is given by:\[T = a(X_{1} + X_{2} + \cdots + X_{n}) + b\]To find its expectation, calculate:\[E[T] = E[a(X_{1} + X_{2} + \cdots + X_{n}) + b] = aE[X_{1} + X_{2} + \cdots + X_{n}] + E[b]\]

Step 3: Apply the Linearity of Expectation

Using the linearity of expectation, we have:\[E[X_{1} + X_{2} + \cdots + X_{n}] = E[X_{1}] + E[X_{2}] + \cdots + E[X_{n}] = n\mu\]Hence,\[E[T] = a(n\mu) + b\]

Step 4: Set Up the Unbiased Condition

Since \( T \) is an unbiased estimator for \( \mu \), we set:\[E[T] = an\mu + b = \mu\]

Step 5: Solve for Constants a and b

From the equation \( an\mu + b = \mu \), we need to find \( a \) and \( b \).Dividing through by \( \mu \), we have:\[an + \frac{b}{\mu} = 1\]Solving for \( a \) and \( b \), come up with two conditions:1. \( an = 1 \rightarrow a = \frac{1}{n} \)2. \( b = 0 \)Thus, the constants \( a \) and \( b \) are \( a = \frac{1}{n} \) and \( b = 0 \).

Key Concepts

Expectation of a Random Variable

The expectation of a random variable, often denoted as \( E[X] \), is a measure that gives a comprehensive view of where the average values of a random variable lie. It is a fundamental concept in statistics and probability theory. Essentially, it works like a "weighted average" of all possible values a random variable can take, where each value is weighted by its probability of occurring. For a discrete random variable, it's calculated as:\[E[X] = \sum_{i} x_i P(x_i)\]where \( x_i \) represents the possible values that the random variable can take, and \( P(x_i) \) is the probability of \( x_i \). For a continuous random variable, integrals replace summations.

When working with random variables like \( X_1, X_2, \ldots, X_n \) in the exercise, the expectation of each \( X_i \) is given by \( \mu \), indicating that each random variable has the same expected value. This uniform expectation across variables simplifies calculations and understanding of complex statistical operations such as sums or linear combinations of these variables.

Linearity of Expectation

A powerful property in statistics and probability theory is the linearity of expectation. It states that the expected value of the sum of random variables is equal to the sum of their expected values. This property holds true regardless of whether the variables are independent or not. Mathematically, for random variables \( X_1, X_2, \ldots, X_n \), the linearity principle can be expressed as:\[E[X_1 + X_2 + \cdots + X_n] = E[X_1] + E[X_2] + \cdots + E[X_n]\]This principle is particularly handy in simplifying calculations in probability.

For example, in the exercise, this rule helps us determine the expectation of \( T = a(X_1 + X_2 + \cdots + X_n) + b \) by transforming it into a simpler form. With provided expectations \( E[X_1] = E[X_2] = \cdots = E[X_n] = \mu \), using the linearity, it follows that:\[E[a(X_1 + X_2 + \cdots + X_n) + b] = a(n\mu) + b\]This realization highlights how linearity propels us toward finding the unbiased conditions for estimation, making it easier to solve for constants like \( a \) and \( b \).

Estimation Theory

Estimation theory is a branch of statistics that deals with the determination of the underlying parameters of a statistical model using sample data. A central concept in estimation theory is the idea of an unbiased estimator. An estimator is unbiased if its expected value equals the true value of the parameter it aims to estimate.

For instance, in the problem, the formula \( T = a(X_1 + X_2 + \cdots + X_n) + b \) is used to estimate the parameter \( \mu \). For \( T \) to be an unbiased estimator of \( \mu \), the condition \( E[T] = \mu \) must be satisfied. By solving this equation under the unbiased condition, we derive that \( a = \frac{1}{n} \) and \( b = 0 \), key values ensuring our estimation does not systematically misrepresent \( \mu \).

In practice, finding an unbiased estimator is crucial for precision and accuracy in statistical analysis, ensuring that estimates do not lean consistently above or below the true value over repeated sampling.