Convergence of uniform, independent RV's

**Maccaman** · August 22nd 2009, 08:08 PM

$X_1, X_2,.....$ are independent random variables that all have a uniform distribution, U(0,1).

Prove that if $M_n = max (X_1, X_2, ..., X_n)$ then as n approaches infinity, $n(1-M_n)$ converges in distribution to an Exp(1) random variable.

I've done a little bit but I have confused myself and I don't know what to do with it now (and im not even sure if Im on the right track

).

So

$Pr(n(1-M_n) \leq x) = Pr(1 - \frac{x}{n} \leq M_n)$

$= 1 - Pr(M_n < 1 - \frac{x}{n})$

$= 1 - Pr(\forall n, X_n < 1 - \frac{x}{n})$

$= 1 - \prod_{i=1}^n \ Pr(X_i < 1 - \frac{x}{n})$

$= 1 - (1-\frac{x}{n})^n$

and this is where I get lost. I think maybe.....

$= 1 - exp(n \ln (1 - \frac{x}{n}))$

but then where to go from here I dont know, which makes me doubt the accuracy of what I have done. Thanks for any help you can provide me with.

**matheagle** · August 22nd 2009, 08:14 PM

you did the hard work, you just need to review calc 2

$\biggl(1-\frac{x}{n}\biggr)^n\to e^{-x}$ done

**Maccaman** · August 22nd 2009, 08:41 PM

Ahhh, off course.

Why I overcomplicate things sometimes I'll never know

**matheagle** · August 22nd 2009, 08:44 PM

Don't be so hard on yourself.
We have Moo at this forum for that.

Thread: Convergence of uniform, independent RV's

LinkBack

Thread Tools

Display

Convergence of uniform, independent RV's

Similar Math Help Forum Discussions

Sum of Two Independent Random Variables (uniform)

Show uniform convergence implies equicontinuity and uniform boundedness

Independent Uniform Random Variables

Sum of 2 independent uniform r.v.

Uniform Continuous and Uniform Convergence

Search Tags