Problem 4 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Let \(X\) be a random variable with \(\mathrm{E}[X]=2\), \(\operatorname{Var}(X)=4\). Compute the expectation and variance of \(3-2 X\).

Step-by-Step Solution

Answer

Expectation is -1, and variance is 16.

Step 1: Identify the Problem

We need to find the expectation and variance of a linear transformation of the random variable \(X\), specifically \(Y = 3 - 2X\).

Step 2: Use Expectation Property

To find the expectation \(\mathrm{E}[Y]\), use the linearity property of expectation. For a linear transformation \(Y = aX + b\), the expectation is given by \(\mathrm{E}[Y] = a\mathrm{E}[X] + b\). Here, \(a = -2\) and \(b = 3\). Thus, compute \(\mathrm{E}[Y] = -2 \times 2 + 3 = -4 + 3 = -1\).

Step 3: Use Variance Property

To find the variance \(\operatorname{Var}(Y)\), use the property of variance for a linear transformation, \(\operatorname{Var}(Y) = a^2 \operatorname{Var}(X)\) when \(Y = aX + b\). In this problem, \(a = -2\), so \(\operatorname{Var}(Y) = (-2)^2 \times 4 = 4 \times 4 = 16\).

Step 4: Combine Results

Conclude the results using the calculations from previous steps. The expectation \(\mathrm{E}[3 - 2X]\) is \(-1\) and the variance \(\operatorname{Var}(3 - 2X)\) is \(16\).

Key Concepts

Linear Transformation

Linear transformation is a fundamental concept when dealing with random variables. It involves reshaping or adjusting a random variable using a linear equation. In simple terms, a linear transformation has the form \( Y = aX + b \), where \( a \) and \( b \) are constants.
This equation alters the random variable \( X \) by scaling it with \( a \) and then shifting it by \( b \). Let's break this down with our example.

• Given \( Y = 3 - 2X \), the linear transformation here involves a scaling factor of \( a = -2 \) and a shift of \( b = 3 \).
• The transformation gives us a new variable \( Y \), derived from \( X \), which impacts both expectation and variance.
• It is crucial to apply the correct mathematical properties to find the expected results for these new transformations.

Understanding linear transformations helps in reshaping data, predicting outcomes with adjustments, or comparing variables under different scales.

Expectation

Expectation, often represented as \( \mathrm{E}[X] \), describes the average or mean value of a random variable. It's like the practical center of a probability distribution, indicating what you'd expect to get, on average, if an experiment were performed many times.
In context of linear transformations, the expectation is calculated using a specific property of expectation:

• For a transformed variable \( Y = aX + b \), the new expectation is \( \mathrm{E}[Y] = a\mathrm{E}[X] + b \).
• In our problem, substituting \( a = -2 \) and \( b = 3 \), and knowing that \( \mathrm{E}[X] = 2 \), we find \( \mathrm{E}[Y] = -2 \times 2 + 3 = -1 \).

This result indicates that the average or "expected" value of the new variable \( Y = 3 - 2X \) is \(-1\).
This principle is immensely helpful as it allows us to understand how shifts and scales alter the expected outcomes of transformations.

Variance

Variance is a measure of the spread, or how much the values of a random variable deviate from the mean. It's represented by \( \operatorname{Var}(X) \). Variance is crucial because it provides insight into the consistency of the data - a small variance means data points are near the mean, while a large variance shows more spread out values.
For linear transformations, variance involves a notable property:

• For a transformed variable \( Y = aX + b \), variance is given by \( \operatorname{Var}(Y) = a^2 \operatorname{Var}(X) \).
• In the exercise, with \( a = -2 \) and \( \operatorname{Var}(X) = 4 \), the new variance becomes \( \operatorname{Var}(Y) = (-2)^2 \times 4 = 16 \).

This calculation shows how scaling affects variability - the variance of \( Y \) (or \( 3 - 2X \)) is \( 16 \), demonstrating increased spread due to scaling by \( a = -2 \).
Variance’s role in linear transformations is pivotal as it helps us understand how changes in scale affect the variability of the new distribution.