statistics?

Question

let bar of xn and s^2 denote the sample mean and variance for the sample x1,...., xn and let bar of xn+1 and s^2n+1 denote these quantities when an additional observation xn+1 is added to the sample.
a) show how bar of xn+1 be computed from bar of xn and xn+1.
b) show that ns^2n+1=(n-1)s^2n + (n/(n+1))(Xn+1- bar of Xn)^2 so that s^2n+1 can be compared from xn+1, bar of Xn and S^2n

Answer 1

a) bar x,n= (x,1 + ... + x,n)/n

bar x,(n+1) = (x,1+ ... + x,n + x,(n+1))/(n+1)
... = (x,1 + ... + x,n)/(n+1) + x,(n+1)/(n+1)
... = [n/(n+1)](bar x,n) + x,(n+1)/(n+1)

Answer 2

Okay, I'm very tired of reading "bar of xn" and "s^2n+1", both of which are confusing, so I'm going to use m for the mean of the first n items, M for the mean of the first n+1 items, s² and S² for the variances of the first n and n+1 items, respectively, x for any sample up to the nth (this will only appear in summation operators, so it's unnecessary to specify a specific sample), and X for the n+1th sample. This will make this response easier to read. Now:

#1: m=(∑x)/n and M=((∑x)+X)/(n+1). Since ∑x=nm, we have that M=((nm)+X)/(n+1)

#2: s²=(∑(x-m)²)/n. Expanding these terms, we get s²=(∑(x²-2xm+m²))/n. Using linearity of summation, this becomes s²=(∑x²)/n - 2m(∑x)/n + m²(∑1)/n. Note that (∑x)/n=m and ∑1=n (1 is added once for each term, and there are n terms), so this is s²=(∑x²)/n-2m²+m²=(∑x²)/n-m². From this formula, S²=((∑x²)+X²)/(n+1)-M². But ∑x²=n(s²-m²), thus we derive the formula S²=(n(s²-m²)+X²)/(n+1) - M². Since M is computable from X and m, it follows that S² is computable from X, m, and s². Proving the equivalence of this to the formula the teacher gave you (which frankly is nearly unreadable) is left as an exercise to the reader.

Answer 3

for readability:
let m(n) = bar of xn
let m(n+1) = bar of xn+1
then m(n+1) = ( n * m(n) + x(n+1) ) / (n+1)

The derivation of the change to the variance eludes me right now, but I recall it being nearly as simple a formula as that shown above for the mean.

Answer 4

i'll say b?
я скажу Б?

Answer 5

ok i give up, what language is that?