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Chapter 10 Question Page 6-7

The integral is shown to be equal to the sum of 2 independent random variables with distributions N(0, 0.5) and N(0, 2). You know they are normally distributed because that's in the definition of the Wiener process, and you know they are independent because that's the independent increments property of the Wiener process, also in the definition of the Wiener process (i.e. how it behaves between time 1.5 and 2 is independent of how it behaves between time 2 and 2.5).

The sum of independent normally distributed random variables is also a random variable with a normal distribution, see here: https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

i.e. X~N(0, 0.5) and Y~N(0, 2), so X+Y~N(0, 2.5)
 
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