Understanding Linearity in Expectation: A Practical Example

Introduction to Linearity of Expectation

In probability theory and statistics, the linearity of expectation is a fundamental property that allows us to break down complex expectations into simpler components. Linearity of expectation states that for any two random variables X and Y, and any constants a and b, the expected value of the sum or linear combination of these random variables is equal to the sum or linear combination of their expected values. Mathematically, this can be expressed as:

Linearity of Expectation:

E(aX bY) aE(X) bE(Y)

This property holds true even if the random variables are not independent, which is a significant simplification over other properties like the law of total expectation.

Applying Linearity of Expectation

Let's apply the concept of linearity of expectation to solve the given problem:

Given Problem:

EA 75 EB 13 c 15

The problem requires us to find EcA * B.

To solve this, we use the linearity of expectation property and the given expression:

EcA * B c * EA * c * EB

Substituting the given values into the equation:

EcA * B 15 * 75 * 15 * 13 1320

By factoring out the constant c, we can break down the problem into simpler parts and then combine the results, which is a practical application of the linearity of expectation.

Deriving the Expression Step-by-Step

To better understand how the linearity of expectation works in this specific example, let's break down the steps in a more detailed manner:

Step 1: Start with the given values:

EA 75 EB 13 c 15

Step 2: Write the expression for EcA * B:

EcA * B c * EA * c * EB

Step 3: Substitute the given values into the expression:

EcA * B 15 * 75 * 15 * 13

Step 4: Perform the multiplication:

EcA * B 15 * 75 1125

Step 5: Multiply the result by c (15) and EB (13):

EcA * B 1125 * 13 1320

Conclusion

The linearity of expectation simplifies complex problems by breaking them down into more manageable parts and then combining the results. In this example, we used this property to find the value of EcA * B by first multiplying the constants and then applying the linearity of expectation. Understanding and applying the linearity of expectation is crucial in various fields, including statistics, probability theory, and theoretical computer science. By mastering this concept, one can tackle more complex problems with ease.

Further Reading and Resources

For those who want to delve deeper into the concept of linearity of expectation, there are several resources available. Books like "Introduction to Probability" by Joseph K. Blitzstein and Jessica Hwang provide comprehensive explanations with various examples. Online platforms such as Khan Academy and Coursera also offer excellent courses on probability and statistics where the linearity of expectation is covered in detail. Familiarizing yourself with these resources can help solidify your understanding and application of this important property.