Let the population be x, and extract n i. i .d .X 1, X2, ..., Xn, and the average sample value is y = (x1+x2+...+xn)/n.
Sample variance is s = ((y-x1) 2+(y-x2) 2+...+(y-xn) 2)/(n-1).
For the convenience of labeling, we only look at the molecular part of S, and set it as A, then EA = E (n * y 2-2 * y * (x1+x2+...+xn)+(x12+x2+...+xn2)) =
Note that EX 1 = EX2= EXn = EY = EX.
VarX 1 = VarX2 =？ e(x^2)(ex)^2。 )
VarY = VarX / n .
So EA = n (varx+(ex) 2)-n * (vary+(ey) 2) = n (varx+(ex) 2)-n * (varx/n+(ex) 2) = (n-1) varx. Get a license.
Explanation:
1. In the probability distribution, let x be a discrete random variable. If E {[x-E(X)] 2} exists, then E {[x-e (x)] 2} is called the variance of x, which is denoted as D(X), Var(X) or DX, where e (.
2. The square root is a concave function, so the negative deviation (through Jensen inequality) is introduced, and the negative deviation depends on the distribution, so the standard deviation of the corrected sample (using Bessel correction) is biased. Unbiased estimation of standard deviation is a technical problem, although the term n- 1 is used. The normal distribution of 5 forms an unbiased estimate.
3. It is necessary to study the deviation of random variables from their mean. So, what kind of quantity is used to measure this deviation? It is easy to see that E[|X-E[X]|] can measure the deviation of a random variable from its mean value E(X). However, due to the absolute value of the above formula, the operation is inconvenient. Generally, the numerical characteristic of dose E[(X-E[X])2] is variance.