Multiple random variables

Ques: If there are 'n' independent random variables i.e. X1, X2,......,Xn; each having zero mean and variance=1. If their sum is defined as Sn=X1+X2+X3+......+Xn. Then, answer the following:

(a) Find and comment on the mean and variance of (Sn/n) as n -> ∞.
(b) Find and comment on the mean and variance of (Sn/√n) as n -> ∞.

Please give the detailed explanation for the answer and also suggest some good study material on the random variables and processes.

Thanks in advance.

It's been a while since I studied probability, but I do remember a few things about the mean of sum of independent random variables.

For a sum of independent random variables, the resultant mean is simply a sum of the individual mean values. Since all values are zero in our case, I would say that the mean of S_n/n and of
S_n/√n will be zero.

The variance of the sum is also a sum of the individual variances. For S_n/n, the variance is easy enough to figure out: since each variance equals 1, we can write the numerator term as n*1 because we would be adding 1 'n' times; when this is divided by n, the answers equals 1. Hence the variance of S_n/n will be equal to 1.

For the case of S_n/√n, you will need to simply calculate the following: \[\lim_{n \to \infty}\]\[\frac{n*1}{sqrt{n}\]

The inner expression will reduce to sqrt. And applying limits to that will give us the answer \[\infty\].

I hope I'm right.

Helpful source: https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

Thanks for help.

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Thanks for help.