Statistics problem

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I haven't had statistics in almost 3 years, and I've encountered a problem that goes as follows:



The book used in this course is completely rubbish. So how do I solve this?

I am presuming this N(1,2) notation refers to mu=1 and sigma^2=2, which appears consistent based on the fact the problem works.

First for the mean:
You will have t terms of 1 divided by t, so therefore the mean of the running mean will be 1.

Next for the variance:
Since variance of a sum of random variables in the sum of the individual variances, the summation alone will have a variance of 2*t. However, multiplying a random variable by a constant linearly changes the standard deviation, dividing the distribution by t will actually divide the variance by t^2. Therefore the net result is 2/t.

Thus the distribution is equal to N(1,2/t).