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## Homework Statement

I have B

_{t+s}~ N (0, t+s) and B

_{s}~ N(0,s).

I want to find the mean and variance of B

_{t+s}+ B

_{t+s}- B

_{s}- B

_{s}.

## Homework Equations

(i) I know that the sum of independent normals is again normal with mean = sum of means and variance = sum of variances.

(ii) I also know that cX ~ N(cμ, c

^{2}σ

^{2})

## The Attempt at a Solution

Obviously the mean is easy to calculate since it is 0 all round, but I'm having problems with the variance.

If I wanted to find distribution of:

B

_{t+s}+ B

_{t+s}it would be ~N(0,t+s+t+s)

No issues here

But now since I have a negative infront of the B

_{s}, how will that come into effect? Notice how property (i) above is regarding the sum of independent normals, what about the difference between two independent normals?

Do I change -B

_{s}(by property (ii) above) so that it has a distribution ~ N ((-1)0, (-1)

^{2}(s))

Does this change the negative out the front of B

_{s}to a positive?

i.e, -B

_{s}~ N (0, s) and then sum the variances so that:

B

_{t+s}+ B

_{t+s}- B

_{s}- B

_{s}~ N (0, (t+s)+(t+s)+(s)+(s))

i.e. B

_{t+s}+ B

_{t+s}- B

_{s}- B

_{s}~ N (0, 2t+4s)?

hope it all makes sense

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