Convergence in probability distribution

Click For Summary

Homework Help Overview

The discussion revolves around a problem in probability theory concerning the convergence of a sequence of random variables. Specifically, the original poster is tasked with showing that the normalized random variable \(\frac{X_n}{n}\), where \(X_n\) follows a geometric distribution, converges in distribution to an exponential distribution as \(n\) approaches infinity.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants express uncertainty about the necessary laws or theorems required to demonstrate the convergence. There are questions about the definition of convergence in distribution and the specific distributions involved. Some participants suggest that understanding the foundational concepts is crucial for progressing with the problem.

Discussion Status

The discussion is ongoing, with participants seeking clarification on key concepts and definitions. There is an acknowledgment of the need for foundational knowledge to tackle the problem effectively. Some guidance has been offered regarding the importance of understanding convergence in distribution.

Contextual Notes

Participants mention that the course literature does not provide sufficient examples related to this topic, which contributes to their difficulties in understanding the problem. There is also a reference to the need for background knowledge from earlier courses.

Elekko
Messages
14
Reaction score
0

Homework Statement


Let X_n \in Ge(\lambda/(n+\lambda)) \lambda>0. (geometric distribution)
Show that \frac{X_n}{n} converges in distribution to Exp(\frac{1}{\lambda})

Homework Equations


I was wondering if some kind of law is required to use here, but I don't know what
Does anyone know how this actually can be shown?
I'm taking a probability class, but the course literature I'm using is does not actually cover examples in this part at all. So it makes me hard to understand this :(
 
Last edited:
Physics news on Phys.org
Elekko said:

Homework Statement


Let X_n \in Ge(\lambda/(n+\lambda)) \lambda>0.
Show that \frac{X_n}{n} converges in distribution to Exp(\frac{1}{\lambda})

Homework Equations


I was wondering if some kind of law is required to use here, but I don't know what
Does anyone know how this actually can be shown?
I'm taking a probability class, but the course literature I'm using is does not actually cover examples in this part at all. So it makes me hard to understand this :(

What is "Ge(.)"?
 
Ray Vickson said:
What is "Ge(.)"?

Geometric distribution
 
Elekko said:

Homework Statement


Let X_n \in Ge(\lambda/(n+\lambda)) \lambda>0. (geometric distribution)
Show that \frac{X_n}{n} converges in distribution to Exp(\frac{1}{\lambda})

Homework Equations


I was wondering if some kind of law is required to use here, but I don't know what
Does anyone know how this actually can be shown?
I'm taking a probability class, but the course literature I'm using is does not actually cover examples in this part at all. So it makes me hard to understand this :(

(1) What is the definition of convergence in distribution? (If you do not know or understand this you cannot profitably proceed further.)

(2) Assuming you have answered (1) correctly, just write down the actual quantities involved (distributions, etc.) and look at what happens when n → ∞. (You should find this to be straightforward; if not, you need to go back to some earlier courses to fill in some missing background.)
 

Similar threads

  • · Replies 6 ·
Replies
6
Views
2K
Replies
2
Views
1K
Replies
2
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 8 ·
Replies
8
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
Replies
3
Views
1K