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Consider the sample standard deviation s=sqrt(1/Nsum_(i=1)^N(x_i-x^_)^2) (1) for n samples taken from a population with a normal distribution. The distribution of s is then ...

Probability and Statistics

A sufficient condition on the Lindeberg-Feller central limit theorem. Given random variates X_1, X_2, ..., let <X_i>=0, the variance sigma_i^2 of X_i be finite, and variance ...

The median of a statistical distribution with distribution function D(x) is the value x such D(x)=1/2. For a symmetric distribution, it is therefore equal to the mean. Given ...

alpha=1/Nsum_(i=1)^N|x_i-mu|=<|x_i-mu|>.

A set of statistical distributions having different variances.

A set of statistical distributions having the same variance.

n sets of s trials each, with the probability of success p constant in each set. var(x/n)=spq+s(s-1)sigma_p^2, where sigma_p^2 is the variance of p_i.

If the fourth moment mu_4!=0, then P(|x^_-mu_4|>=lambda)<=(mu_4+3(N-1)sigma^4)/(N^3lambda^4), where sigma^2 is the variance.

(Deltau)_i^2=(u_i-u^_)^2, where u^_ is the average of {u_i}.