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February 24th, 2017, 07:21 PM  #1 
Newbie Joined: Aug 2013 Posts: 18 Thanks: 0  (Brownian Motion) Proving that a random variable X is normal?
I have attached the question below! So given that W(t) is a Brownian motion, how do you prove X=W(a)+W(b) is normal? I know it must satisfy this condition: that W(t)W(s)~N(0,ts). My initial thought process: X(t)X(s)=(W(t)+W(t))(W(s)+W(s))=2(W(t)W(s)), but since the interval is 0<a<b, I'm pretty sure the variable substitutions I've done is incorrect. Is my approach to this problem incorrect? Are the W(a)+W(b) just constants? But that would entail that (W(a)+W(b))(W(a)+W(b))=0? Sorry, I don't think I am grasping the concepts very well. I would appreciate any pointers on how to solve this problem! Last edited by facebook; February 24th, 2017 at 07:39 PM. 
February 24th, 2017, 07:31 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,009 Thanks: 1042 
how are we supposed to read that image?

February 24th, 2017, 07:38 PM  #3 
Newbie Joined: Aug 2013 Posts: 18 Thanks: 0 
Hi, sorry I've corrected the image! The original file got downscaled, I did not anticipate image compression, my apologies

February 26th, 2017, 02:33 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,540 Thanks: 591 
You must show W(a)+W(b)W(ac)W(bc) is normal. The mean = 0, but I am not sure what the variance is.


Tags 
brownian, motion, normal, proving, random, variable, variance 
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