Understanding and Calculating the Expected Value for Linear Combinations of Random Variables

In the field of probability and statistics, understanding the expected value of linear combinations of random variables is a fundamental concept. This article aims to provide a comprehensive guide on how to calculate the expected value of a linear combination like E(5X - 9) when given the expected values and variances of X and Y.

Introduction to Expected Values

The expected value, often denoted as E(X), is a measure that and quantifies the long-run average result of repetitions of the same experiment it represents. It can be defined as the sum of all possible values multiplied by the probability of occurrence of each value. For a discrete random variable, this is given as:

E(X) Σ xiP(X xi)

Expected Values for Linear Combinations of Random Variables

In practical applications, it is often necessary to work with linear combinations of random variables. The expected value of such combinations can be calculated using the linearity property of expected values. The property states that for any two random variables X and Y and any constants a, b, c, and d, the following holds:

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

This linearity property makes it straightforward to compute the expected values of more complex expressions.

Calculating E(5X - 9)

Let's consider a specific example where X and Y are independent random variables with given expected values:

E(X) 8 E(Y) 12

We need to calculate the expected value of the expression 5X - 9. Using the linearity property of expected values, we can break this down into simpler steps:

E(5X - 9) E(5X) - 9

Since the expectation of a constant is the constant itself, we have:

E(5x) 5E(X)

Using the given expected value E(X) 8, we can now compute:

E(5X) 5 * 8 40

Therefore:

E(5X - 9) 40 - 9 31

Mathematical Proof and Verification

To verify the result, let's write out the steps again in a clear mathematical form:

E(5X - 9) E(5X) - E(9)

E(5X) 5E(X) 5 * 8 40

E(9) 9 (since the expectation of a constant is the constant itself)

E(5X - 9) 40 - 9 31

Conclusion and Additional Insights

The calculation above demonstrates the utility and power of the linearity property of expected values. It simplifies the process of computing expected values for linear combinations of random variables, which is a common task in many statistical analyses and modeling scenarios.

Understanding and applying these principles can save time and effort in complex probabilistic calculations, making it an essential skill for statisticians, data scientists, and anyone working with random variables and their expected values.

Related Topics and References

Expected Value of a Function of a Random Variable: This involves understanding how to compute the expected value when a function of a random variable is considered. Covariance and Correlation: These concepts are closely related to the expected values and can be useful in understanding the relationships between random variables. Variance and Standard Deviation: These measures provide insights into the spread and variability of a random variable, complementing the concept of expected value.