SUMMARY
The expected value of a bivariate probability density function (pdf) fXY(x,y) is calculated using double integrals. Specifically, for the function f(x,y) = x + y, the expected value E[X] is determined by the double integral E[X] = ∫∫ x * (x + y) dx dy over the limits 0 < x < 1 and 0 < y < 1. The result of this calculation yields E[X] = 7/12. This confirms that a single integration only provides the conditional expectation of one variable with respect to the other.
PREREQUISITES
- Understanding of bivariate probability density functions
- Knowledge of double integrals in calculus
- Familiarity with the concept of expected value in statistics
- Basic proficiency in mathematical notation and integration techniques
NEXT STEPS
- Study the properties of bivariate probability density functions
- Learn how to perform double integrals in various coordinate systems
- Explore conditional expectations and their applications in statistics
- Investigate the implications of expected values in real-world scenarios
USEFUL FOR
Mathematicians, statisticians, and students studying probability theory, particularly those interested in understanding the computation of expected values in bivariate distributions.