from each other: Absolutely continuous random variable If the random variable has a probability density function , and is the corresponding cumulative distribution function, then or equivalently, where is the expected value of given by In these formulas, the integrals with respect to and are Lebesgue and Lebesgue–Stieltjes integrals, respectively. If the function is Riemann-integrable on every finite interval then where the integral is an improper Riemann integral. Examples Exponential distribution The exponential distribution with parameter is a continuous distribution whose probability density function is given by on the interval . Its mean can be shown to be Using integration by parts and making use of the expected value already calculated, we have: Thus, the variance of is given by Fair die A fair six-sided die can be modeled as a discrete random variable, , with outcomes 1 through 6, each with equal probability 1/6. The expected value of is Therefore, the variance of is The general formula for the variance of the outcome, , of an die is Commonly used probability distributions The following table lists the variance for some commonly used probability distributions. Properties Basic properties Variance is non-negative because the squares are positive or zero: The variance of a constant is zero. Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value: Issues of finiteness If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index satisfies Decomposition The general formula for variance decomposition or the law of total variance is: If and are two random variables, and the variance of exists, then The conditional expectation of given , and the conditional variance may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation given the event Y = y. This quantity depends on the particular value y; it is a function . That same function evaluated at the random variable Y is the conditional expectation In particular, if is a discrete random variable assuming possible values with corresponding probabilities , then in the formula for total variance, the first term on the right-hand side becomes