Are min(X,Y) and X-Y independent given X>Y in an exponential distribution?

  • Thread starter Thread starter Chris L T521
  • Start date Start date
Click For Summary
SUMMARY

The discussion centers on the independence of the random variables min(X,Y) and X-Y, given that X > Y, where X and Y are independent exponential random variables with rates λ and μ. The problem requires proving that these two variables are independent under the specified condition. Although no solutions were provided in the forum, the topic is critical for understanding properties of exponential distributions in probability theory.

PREREQUISITES
  • Understanding of exponential random variables and their properties
  • Knowledge of conditional probability and independence
  • Familiarity with probability density functions (PDFs)
  • Basic skills in mathematical proof techniques
NEXT STEPS
  • Study the properties of independent exponential random variables
  • Learn about conditional independence in probability theory
  • Explore the concept of joint distributions and their implications
  • Investigate applications of exponential distributions in real-world scenarios
USEFUL FOR

Statisticians, data scientists, and students of probability theory who are interested in the behavior of exponential distributions and their applications in statistical modeling.

Chris L T521
Gold Member
MHB
Messages
913
Reaction score
0
Here's this week's problem.

-----

Problem: Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$. Argue that, conditional on $X>Y$, the random variables $\min(X,Y)$ and $X-Y$ are independent.

-----

 
Physics news on Phys.org
No one answered this week's problem. I haven't completed the solution yet (almost done), but I'll update this post later today with a solution. Sorry about that!
 

Similar threads

High School Can x(x+y) Be Proved to Be ≤ 2 for Positive Real Numbers?
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Projection Map $X \times Y$: Closure Property
  • · Replies 1 ·
Replies
1
Views
4K
Undergrad What is the solution to this week's POTW?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Are There a Finite Number of Solutions to $f(x) = y$ in a Bounded Region?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Expected value of X~Geom(p) given X+Y=z
  • · Replies 5 ·
Replies
5
Views
2K
High School How Do You Solve for \(x^4 + y^4\) Given These Equations?
  • · Replies 1 ·
Replies
1
Views
1K
Determine marginal densities and distributions from joint density
  • · Replies 7 ·
Replies
7
Views
1K
High School How do I solve for non-negative integers x and y?
  • · Replies 1 ·
Replies
1
Views
1K
High School What Are the Non-Negative Integer Solutions to (xy-7)² = x² + y²?
  • · Replies 1 ·
Replies
1
Views
2K
Problem of the Week #124 - August 11th, 2014
  • · Replies 1 ·
Replies
1
Views
1K