Convergence in probability distribution

  • Thread starter Elekko
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Homework Statement


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

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 :(
 
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Answers and Replies

  • #2
Ray Vickson
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Homework Statement


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

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(.)"?
 
  • #4
Ray Vickson
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Homework Statement


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

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.)
 

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