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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!
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Last edited by facebook; February 24th, 2017 at 07:39 PM.
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February 24th, 2017, 07:31 PM   #2
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how are we supposed to read that image?
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February 24th, 2017, 07:38 PM   #3
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Hi, sorry I've corrected the image! The original file got downscaled, I did not anticipate image compression, my apologies
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February 26th, 2017, 02:33 PM   #4
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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.
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