Calculating Expected Values for Independent Random Variables in Probability Theory

In probability theory, the expected value (or expectation) of a random variable is a measure of the long-run average value of repetitions of the experiment it represents. This article delves into calculating the expected values for the random variables x and y, where their density functions are given, and they are assumed to be independent. The calculations will also explore the linearity of expectation and the concept of independent random variables in relation to the expected value of their products.

Understanding the Density Functions

First, let's consider the two random variables, x and y, with their respective density functions:

For x: f_X(x) frac{1}{8}x for 0 ≤ x ≤ 4 For y: f_Y(y) frac{1}{12}y for 1 ≤ y ≤ 5

Calculate Ex and Ey

The expected value Ex is calculated as:

For x:

Ex int_{0}^{4} x f_X(x) dx int_{0}^{4} xleft(frac{1}{8}xright) dx frac{1}{8} int_{0}^{4} x^2 dx

Therefore, int_{0}^{4} x^2 dx left[frac{x^3}{3}right]_{0}^{4} frac{4^3}{3} frac{64}{3}

Thus, Ex frac{1}{8} cdot frac{64}{3} frac{8}{3}

For y:

Ey int_{1}^{5} y f_Y(y) dy int_{1}^{5} yleft(frac{1}{12}yright) dy frac{1}{12} int_{1}^{5} y^2 dy

Therefore, int_{1}^{5} y^2 dy left[frac{y^3}{3}right]_{1}^{5} frac{5^3}{3} - frac{1^3}{3} frac{125}{3} - frac{1}{3} frac{124}{3}

Thus, Ey frac{1}{12} cdot frac{124}{3} frac{31}{9}

Calculate E(2x 3y)

Using the linearity of expectation:

E(2x 3y) 2Ex 3Ey

Substituting the values we found:

E(2x 3y) 2 cdot frac{8}{3} 3 cdot frac{31}{9}

To add these fractions, convert frac{8}{3} to a denominator of 9:

frac{8}{3} frac{24}{9}

Thus, E(2x 3y) frac{24}{9} frac{93}{9} frac{141}{9}

Calculate E(xy)

Since x and y are independent:

E(xy) Ex cdot Ey

Substituting the values:

E(xy) frac{8}{3} cdot frac{31}{9} frac{248}{27}

Summary of Findings

E(2x 3y) frac{141}{9} E(xy) frac{248}{27}

Key Concepts to Remember

Linearity of Expectation: The expectation of a linear combination of random variables is the same linear combination of their expectations, regardless of whether the random variables are independent. Independence of Random Variables: If two random variables are independent, the expected value of their product is the product of their expected values.

Further Exploration

This problem explores the fundamental concepts of probability theory and provides a solid foundation for understanding and applying these concepts in more complex scenarios. Whether you are a student or a professional in data science, understanding these calculations is essential for working with random variables and their expected values.