SUMMARY
The marginal probability density function (pdf) of the random variable X, given the joint pdf f(x,y) = 8xy for 0 ≤ x ≤ y ≤ 1, is derived by integrating over the appropriate bounds. The correct marginal pdf f_X(x) is calculated as f_X(x) = ∫(8xy) dy from x to 1, resulting in f_X(x) = 4x(1 - x^2). The integration bounds are crucial, as they define the region where the joint pdf is positive. A graphical representation of the region 0 ≤ x ≤ y ≤ 1 aids in understanding the limits of integration.
PREREQUISITES
- Understanding of joint probability density functions (pdf)
- Knowledge of integration techniques in calculus
- Familiarity with the concept of marginal distributions
- Ability to interpret graphical representations of functions
NEXT STEPS
- Study the derivation of marginal pdfs from joint pdfs in continuous random variables
- Learn about graphical methods for visualizing probability distributions
- Explore the properties of continuous random variables and their distributions
- Investigate applications of marginal pdfs in statistical analysis and probability theory
USEFUL FOR
Students studying probability and statistics, mathematicians focusing on continuous random variables, and anyone involved in statistical modeling and analysis.