1. ## random variable convergence

Wn has density [1+(x/n)]/[1+(0.5n)] for 0<x<1, and 0 otherwise. The random variable W has a uniform distribution on [0,1]. Prove that {Wn} converges in distribution to W.

i have so far that
FWn(w) = 0 for all w

Fw(w)=0 if w<=0, w if 0<w<1, 1 if w>=1

How do I show the convergence?

2. ## Re: random variable convergence

Hint: Use characteristic function. (I guess the convergence is in distribution)

3. ## Re: random variable convergence

I have no idea how to use it...

4. ## Re: random variable convergence

I'm guessing the moment generating function works in this situation? Anywho, show that the MGFs converge and then use the fact that MGFs are unique.

5. ## Re: random variable convergence

ok so this is what i get

mWn(s)=E(e^(sWn))= integral (0 to 1): e^(sWn)*((1+(x/n)))/(1+(1/2n))dWn
which equals (as lim n-> inf.)
integral (0 to 1): e^(Sw)dw

mW(s)=E(e^(Sw))=integral (0 to 1): e^(Sw)dw

is this right so far?

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