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The variance of a random variable is the expected ... as the squared Euclidean distance between the random variable and its mean, or, simply ...
The term "random variable" in statistics is traditionally limited to the real-valued case (=). In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution.
It is also the continuous distribution with the maximum entropy for a specified mean and variance. [18] [19] Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other. [20] [21]
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). [1]
the product of two random variables is a random variable; addition and multiplication of random variables are both commutative; and; there is a notion of conjugation of random variables, satisfying (XY) * = Y * X * and X ** = X for all random variables X,Y and coinciding with complex conjugation if X is a constant.
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:
Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. [2] [3] Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values.
Let X (integrable) be a random variable with finite non-zero variance σ 2 (and thus finite expected value μ). [9] Then for any real number k > 0, (| |).Only the case > is useful. . When the right-hand side and the inequality is trivial as all probabilities are ≤