SUMMARY
The statement "If X1 ~ Y1 and X2 ~ Y2, then X1X2 ~ Y1Y2" is proven false through a counterexample. Specifically, if X1 is a random variable with P(X1 = 1) = P(X1 = -1) = 0.5, and X2 is defined as -X1 while Y1 and Y2 are both equal to X1, then X1X2 results in -1 almost surely, whereas Y1Y2 results in 1. This demonstrates that the distributions of the products do not necessarily align even when the individual distributions are the same.
PREREQUISITES
- Understanding of random variables and their distributions
- Familiarity with probability theory concepts
- Knowledge of uniform and binomial distributions
- Basic algebraic manipulation of random variables
NEXT STEPS
- Study the properties of random variable products in probability theory
- Explore counterexamples in statistical proofs
- Learn about the implications of distribution equivalence in random variables
- Investigate the behavior of combined distributions in different scenarios
USEFUL FOR
Students of probability theory, statisticians, and mathematicians interested in understanding the relationships between random variables and their distributions.