Problem 30 - A First Course in Probability

Question

If \(X\) and \(Y\) are independent and identically distributed with mean \(\mu\) and variance \(\sigma^{2},\) find $$ E\left[(X-Y)^{2}\right] $$

Step-by-Step Solution

Answer

The expected value of \((X-Y)^2\) is \(2\sigma^2\).

Step 1: Simplify the expression

Let's first expand the expression \((X-Y)^2\): \((X-Y)^2 = (X-Y)(X-Y) = X^2 - 2XY + Y^2\) Now, our task is to find the expected value of this expression.

Step 2: Apply the properties of expected value

Recall that the expected value operator is linear, i.e., for any random variables \(A\) and \(B\), and any constants \(a\) and \(b\): \(E[aA + bB] = aE[A] + bE[B]\) Thus, we can compute the expected value of the expression \(X^2 - 2XY + Y^2\) as follows: \(E[(X-Y)^2] = E[X^2 - 2XY + Y^2] = E[X^2] - 2E[XY] + E[Y^2]\)

Step 3: Use the properties of independence

Since \(X\) and \(Y\) are independent random variables, then the expected value of their product is the product of their expected values: \(E[XY] = E[X]E[Y] = \mu\mu = \mu^2\) Additionally, we can use the distributive property of variance to find the second moment of \(X\) and \(Y\): \(E[X^2] = Var[X] + (E[X])^2 = \sigma^2 + \mu^2\) Since \(X\) and \(Y\) are identically distributed, the same applies to \(Y\): \(E[Y^2] = Var[Y] + (E[Y])^2 = \sigma^2 + \mu^2\)

Step 4: Substitute the expressions for the expected values in the main equation

Now that we have computed the individual expected values, we can substitute them back into our expression for the expected value of \((X-Y)^2\): \(E[(X-Y)^2] = E[X^2] - 2E[XY] + E[Y^2] = (\sigma^2 + \mu^2) - 2\mu^2 + (\sigma^2 + \mu^2)\)

Step 5: Simplify the expression and state the answer

Finally, combining the terms, we obtain the expected value of \((X-Y)^2\): \(E[(X-Y)^2] = 2\sigma^2 + 2\mu^2 - 2\mu^2 = 2\sigma^2\) Therefore, the expected value of \((X-Y)^2\) is \(2\sigma^2\).

Key Concepts

Independent Random Variables

When we talk about independent random variables, we are focusing on the concept that the outcome of one random variable does not affect the outcome of another.
For example, consider two dice rolls. The result of the first roll doesn't impact the result of the second.
Mathematically, if two random variables \( X \) and \( Y \) are independent, their joint probability is the product of their individual probabilities.

• For expected values, if \( E[X] \) is the expected value of \( X \) and \( E[Y] \) is for \( Y \), then \( E[XY] = E[X]E[Y] \) for independent variables.
• This property simplifies many calculations, like in our exercise where \( E[XY] = \mu^2 \) since both \( X \) and \( Y \) share the same mean, \( \mu \).

Being independent reduces complexity in statistical calculations, making it easier to predict outcomes.

Variance

Variance is a measure of how much a set of numbers, like random variables, differ from their expected value, or mean.
It tells us about the spread or distribution of the data.
The variance of a random variable \( X \), denoted as \( \text{Var}[X] \), is calculated by the equation:\[ \text{Var}[X] = E[X^2] - (E[X])^2 \] This equation shows that variance relates to the second moment around the mean.

• For independent and identically distributed variables \( X \) and \( Y \), with the same variance \( \sigma^2 \), we have \( \text{Var}[X] = \text{Var}[Y] = \sigma^2 \).
• The variance indicates that even if the averages are the same, the random variables might behave differently.

Understanding variance is crucial for predicting and modeling real-world datasets.

Identically Distributed

When random variables are identically distributed, it means they follow the same probability distribution. Each variable, \( X \) and \( Y \), has the same shape, spread, and average.
They share characteristics such as having the same mean \( \mu \) and variance \( \sigma^2 \).

• In our exercise, because \( X \) and \( Y \) are identically distributed, they both have an expected value of \( \mu \) and variance of \( \sigma^2 \).
• This commonality simplifies calculations significantly, as we don’t need separate computations for each variable.

This concept is essential for calculating the collective behaviors of multiple random variables.

Second Moment

The concept of the second moment involves squaring the random variable and finding its expected value.
The second moment about the origin for a random variable \( X \), is expressed as \( E[X^2] \).
It provides insight into the distribution’s spread.

• The second moment is part of calculating variance, since \( \text{Var}[X] = E[X^2] - (E[X])^2 \).
• For identically distributed variables in our problem, \( E[X^2] \) and \( E[Y^2] \) are both equal to \( \sigma^2 + \mu^2 \). This stems from their common distribution characteristics.

Understanding the second moment helps in analyzing variability and understanding the distribution’s shape.