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The expected value of a random variable may be thought of as the value of the arithmetic mean of the observed values as the *sample size increases. It should be noted that in the case of a [[discrete random variable]] the expected value is generally not a possible value. For example, when an unbiased coin is tossed once, the expected value of the number of heads is <math>\frac{1}{2}</math>. | The expected value of a [[random variable]] may be thought of as the value of the arithmetic mean of the observed values as the *sample size increases. It should be noted that in the case of a [[discrete random variable]] the expected value is generally not a possible value. For example, when an unbiased coin is tossed once, the expected value of the number of heads is <math>\frac{1}{2}</math>. | ||
Some random variables (for example, those with a *Cauchy distribution] do not have an expected value. There are also [[discrete random variable]]s without expected values, though this situation can only arise when the set of possible values is infinite and the sum involved is the sum of a series that does not converge. For example, if <math>P(X = r) = K/r^2</math>, for <math>r = 1, 2, ...,</math> where <math>1/K= \sum1/r^2( =\pi^2/6),</math> then the expression for <math>E(X)</math> is <math>\sum K/r</math>, and this series diverges to <math>\infty</math>. For the expected values of sums and products of random variables, see EXPECTATION ALGEBRA. | Some random variables (for example, those with a *Cauchy distribution] do not have an expected value. There are also [[discrete random variable]]s without expected values, though this situation can only arise when the set of possible values is infinite and the sum involved is the sum of a series that does not converge. For example, if <math>P(X = r) = K/r^2</math>, for <math>r = 1, 2, ...,</math> where <math>1/K= \sum1/r^2( =\pi^2/6),</math> then the expression for <math>E(X)</math> is <math>\sum K/r</math>, and this series diverges to <math>\infty</math>. For the expected values of sums and products of random variables, see EXPECTATION ALGEBRA. |