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Measure of dispersion
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- Variance is a measure of dispersion. It is equal to the average squared distance of the realizations of a random variable from its expected value.
www.statlect.com/fundamentals-of-probability/varianceVariance | Definition based on the expected value - Statlect
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Apr 24, 2022 · In both cases, \( f \) is the probability density function of \( X \). Variance is always nonnegative, since it's the expected value of a nonnegative random variable. Moreover, any random variable that really is random (not a constant) will have strictly positive variance.
Variance is a measure of how much a random variable can be expected to vary around its expected value. For example, consider a random variable Y Y representing a fair throw of a six-sided die. The expected value is E[Y] = 3.5 E [Y] = 3.5, but the result may vary between 1 and 6.
Variance. Remember that the variance of any random variable is defined as Var(X) = E [(X − μX)2] = EX2 − (EX)2. So for a continuous random variable, we can write. Also remember that for a, b ∈ R, we always have Var(aX + b) = a2Var(X). (4.4) Example.
Variance reduction. The error in a direct Monte Carlo simulation goes as σ/√n. So there are two ways we can reduce the error. Run the simulation for a longer time, i.e., increase n or find a different formulation of the Monte Carlo that has a smaller σ. Methods that do the latter are know as variance reduction. 5.1 Antithetic variables.
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The variance is the probability weighted average of the square of these variances. The square of the error treats positive and negative variations alike, and it weights large variations more heavily than smaller ones.
Learn how variance is defined in probability theory by using the expected value. Understand the formula that defines variance.
The variance is a measure of how spread out the distribution of a random variable is. Here, the variance of Y is quite small since its distribution is concentrated at a single value, while the variance of X will be larger since its distribution is more spread out.
