My Math Forum (Brownian Motion) Proving that a random variable X is normal?

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February 24th, 2017, 07:21 PM   #1
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(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,t-s).
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!
Attached Images
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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: 2,585 Thanks: 1430 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: 6,835 Thanks: 733 You must show W(a)+W(b)-W(a-c)-W(b-c) 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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