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

  • MHB
  • Thread starter Chris L T521
  • Start date
In summary, an exponential distribution is a probability distribution that describes the time between events in a Poisson process. It is often used to model real-world phenomena, such as the time between calls at a call center or the time until a certain event occurs. Min(X,Y) represents the minimum value between two random variables X and Y, while X-Y represents the difference between those two variables. In order for two random variables to be considered independent in an exponential distribution, the probability of one event occurring must not affect the probability of the other event occurring. This can be represented as P(X and Y) = P(X) * P(Y). It is possible for min(X,Y) and X-Y to be independent even if X is greater than Y
  • #1
Chris L T521
Gold Member
MHB
915
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
  • #2
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!
 

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

What is an exponential distribution?

An exponential distribution is a probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate.

What is the meaning of min(X,Y) and X-Y in the context of an exponential distribution?

min(X,Y) represents the minimum value between two random variables X and Y, while X-Y represents the difference between those two variables.

How do we determine if two random variables are independent in an exponential distribution?

In an exponential distribution, two random variables X and Y are considered independent if the probability of one event occurring does not affect the probability of the other event occurring. This can be mathematically represented as P(X and Y) = P(X) * P(Y).

Can min(X,Y) and X-Y be independent if X>Y in an exponential distribution?

Yes, min(X,Y) and X-Y can be independent in an exponential distribution even if X is greater than Y. This is because the relationship between X and Y is not affected by their order, only by their individual values.

How can we use the exponential distribution to model real-world phenomena?

The exponential distribution has many applications in real-world phenomena, such as modeling the time between calls at a call center, the time between arrivals of customers at a store, or the time between failures of a machine. It can also be used in survival analysis to model the time until a certain event occurs.

Similar threads

MHB Projection Map $X \times Y$: Closure Property
Replies
1
Views
4K
MHB Are There a Finite Number of Solutions to $f(x) = y$ in a Bounded Region?
Replies
1
Views
1K
I Expected value of X~Geom(p) given X+Y=z
Replies
5
Views
1K
MHB Is the adjunction space $X \cup_f Y$ normal for closed $A$ and continuous $f$?
Replies
2
Views
3K
Determine marginal densities and distributions from joint density
Replies
7
Views
817
I The covariance of a sum of two random variables X and Y
Replies
10
Views
2K
MHB How do we prove that $p(x,y)$ is a nonzero element of $\Bbb R[x,y]$?
Replies
1
Views
3K
I Is this conditional expectation identity true?
Replies
1
Views
996
A Proving Lim F(x,y) is the Distribution Function for X
Replies
1
Views
1K
MHB How Do You Prove the Boundedness of an Integral Operator in Measure Spaces?
Replies
1
Views
2K

Recent Insights

Back
Top