SUMMARY
The marginal cumulative distribution function (CDF) of a random variable X, given two random variables X and Y, is defined as F(x) = ∫_{-\infty}^{\infty} F(x|y)f(y)dy. The discussion clarifies that the initial expression F(x) = ∫^{F(x|y)}_{-\infty}f(y)dy is incorrect. The correct formulation requires integrating the conditional CDF F(x|y) multiplied by the probability density function (pdf) f(y). No known distribution validates the initial claim.
PREREQUISITES
- Understanding of probability density functions (pdf)
- Knowledge of conditional probability and conditional pdfs
- Familiarity with cumulative distribution functions (CDF)
- Basic calculus, particularly integration techniques
NEXT STEPS
- Study the derivation of marginal CDFs in multivariate distributions
- Learn about the properties of conditional pdfs and their applications
- Explore examples of joint distributions and their marginal distributions
- Investigate specific distributions that exhibit conditional relationships, such as the bivariate normal distribution
USEFUL FOR
Statisticians, data scientists, and anyone involved in probability theory or statistical modeling who seeks to deepen their understanding of marginal and conditional distributions.