Show sample mean and sample variance are independent
An interesting question is whether, if in a family of distributions the mean determines the variance, the sample mean and sample variance can be independent. The answer is yes: take any family of Normal distributions in which the variance depends on the mean suchProperties of the sample mean and variance Lemma (Facts about chisquared random variables) We use the notation 2 p to denote a chisquared random variable with p degrees of freedom. (a) If Z is a N(0, 1) random variable, then Z2 2 1; that is, the square of a standard normal random variable is a chisquared random variable. show sample mean and sample variance are independent
sample points. Examples are the sample mean x x in and the sample variance s2 (x i x)2n A random sample may be regarded as a microcosm of the population from which it is drawn. Therefore, we might attempt to estimate the moments of the populations p. d. f f(x) by the corresponding moments of the sample statistics.
n is a random sample from a normal distribution with mean, , and variance, 2. It follows that the sample mean, X, is independent of the sample variance, S 2. The sample mean and variance September 12, 2013 117 The sample mean and variance Sample mean and sample variance The gaussian case The 2distribution The distribution of the sample variance The 2density 217 Sample mean and sample variance Let X1, X2, , Xn be independent and identically distributed random variables with expected value andshow sample mean and sample variance are independent Stack Exchange network consists of 175 Q& A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share