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An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications ...
The expectation of a random variable plays an important role in a variety of contexts. ... by means of the computational formula for the variance ...
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
Recall that variance is the expected squared deviation between a random variable (say, Y) and its expected value. The expected value can be thought of as a reasonable prediction of the outcomes of the random experiment (in particular, the expected value is the best constant prediction when predictions are assessed by expected squared prediction ...
Formally, a multivariate random variable is a column vector = (, …,) (or its transpose, which is a row vector) whose components are random variables on the probability space (,,), where is the sample space, is the sigma-algebra (the collection of all events), and is the probability measure (a function returning each event's probability).
For a random variable following the continuous uniform distribution, the expected value is = +, and the variance is = (). For the special case a = − b , {\displaystyle a=-b,} the probability density function of the continuous uniform distribution is:
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of ...
These identities lead to simple calculations of the expected value and variance of any random variable in the exponential family [], []. Expected value of Y : Taking the first derivative with respect to θ {\displaystyle \theta } of the log of the density in the exponential family form described above, we have