- #1

Dassinia

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**Hi, I'm trying to solve this exercise but I really don't know how**

1. Homework Statement

1. Homework Statement

Let X1, X2,.. be a sequence of iid random variables following a uniform distribution on (0,1). Define the random variable N≥2 as the first point in which the sequence (X1,X2,...) stops decreasing. i.e If N=n :

X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gifX2[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gif...[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gifXn-1<Xn

For 0[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gif1 show that

1. P(X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift,N=n)=tn-1/(n-1)! - tn/(n)!

2.P(X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift,N pair)=1-exp(-t) use series expansion of exp(t)+exp(-t) et exp(t)-exp(-t)

3. E[N]=e

## Homework Equations

## The Attempt at a Solution

I solved the third one

For the first one, I think that we have to use induction proof, but I don't see how to do that here ?

Thanks

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