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A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product = is a product distribution.
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.
Despite the newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the Lebesgue integral of X is defined via weighted averages of approximations of X which take on finitely many values ...
The reason the sample covariance matrix has in the denominator rather than is essentially that the population mean is not known and is replaced by the sample mean ¯. If the population mean E ( X ) {\displaystyle \operatorname {E} (\mathbf {X} )} is known, the analogous unbiased estimate is given by
This follows from the definition of independence in probability: the probabilities of two independent events happening, given a model, is the product of the probabilities. This is particularly important when the events are from independent and identically distributed random variables, such as independent observations or sampling with ...
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. [1] There are several kinds of means (or "measures of central tendency") in mathematics, especially in statistics.
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure
In statistics, "random sample" is the typical terminology, but in probability, it is more common to say "IID." Identically distributed means that there are no overall trends — the distribution does not fluctuate and all items in the sample are taken from the same probability distribution.