Search results
Results from the WOW.Com Content Network
For normally distributed random variables inverse-variance weighted averages can also be derived as the maximum likelihood estimate for the true value. Furthermore, from a Bayesian perspective the posterior distribution for the true value given normally distributed observations and a flat prior is a normal distribution with the inverse-variance weighted average as a mean and variance ().
When treating the weights as constants, and having a sample of n observations from uncorrelated random variables, all with the same variance and expectation (as is the case for i.i.d random variables), then the variance of the weighted mean can be estimated as the multiplication of the unweighted variance by Kish's design effect (see proof):
This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y , then the weight of the variance of X will be four times the weight of the variance of Y .
In statistics, pooled variance (also known as combined variance, composite variance, or overall variance, and written ) is a method for estimating variance of several different populations when the mean of each population may be different, but one may assume that the variance of each population is the same.
The expected value of a random variable is the weighted average of the possible values it might take on, with the weights being the respective probabilities. More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable.
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger population of numbers, where "population ...
The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values. If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean. [1]