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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
Expected values can also be used to compute the variance, by means of the computational formula for the variance = [] ( []). A very important application of the expectation value is in the field of quantum mechanics.
In probability theory, the law of total variance [1] or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, [2] states that if and are random variables on the same probability space, and the variance of is finite, then
Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution (,) is the one with maximum entropy. [29] To see this, let X {\textstyle X} be a continuous random variable with probability density f ( x ) {\textstyle f(x)} .
The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations [2] (LIE), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then
In words: the variance of Y is the sum of the expected conditional variance of Y given X and the variance of the conditional expectation of Y given X. The first term captures the variation left after "using X to predict Y", while the second term captures the variation due to the mean of the prediction of Y due to the randomness of X.
Formulas for its mean and variance are in the section #Logarithmic expectation and variance. If X ~ Gamma(α, θ), then follows a generalized gamma distribution with parameters p = 2, d = 2α, and = [citation needed].
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed.Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution.