| is the indicator function of the set A A very important application of the expectation value is in the field of quantum mechanics. ^ What Is the Negative Binomial Distribution? [ Given a discrete random variable X, suppose that it has values x1, x2, x3, . , , , E Intuitively, the expectation of a random variable taking values in a countable set of outcomes is defined analogously as the weighted sum of the outcome values, where the weights correspond to the probabilities of realizing that value. {\displaystyle Y_{0}=X_{1}} p {\displaystyle \mathop {\hbox{E}} } {\displaystyle X} to denote expected value goes back to W. A. Whitworth in 1901. {\displaystyle \mathop {\hbox{E}} } 2 Changing summation order, from row-by-row to column-by-column, gives us. 2 , F {\displaystyle X(\omega )\leq x.} {\displaystyle \langle X\rangle } 0 E , then the expected value is defined as the Lebesgue integral. . [ 1 ∫ The expected value is one such measurement of the center of a probability distribution. X − X values, with the ) X . μ {\displaystyle X^{-}(\omega )\geq -x,} ( , 1 This is saying that the probability mass function for this random variable gives f(xi) = pi. {\displaystyle c} In what follows we will see how to use the formula for expected value. {\displaystyle g:{\mathbb {R} }\to {\mathbb {R} }} x {\displaystyle X} ( i values being the weights. and ) For example, if you were rolling a die, it can only have the set of numbers {1,2,3,4,5,6}. Neither Pascal nor Huygens used the term "expectation" in its modern sense. c ≤ In general, it is not the case that ) x Y x + = = X ( E products ∫ The expected value informs about what to expect in an experiment "in the long run", after many trials. n min for x ⋯ If E 1 In this example, we see that, in the long run, we will average a total of 1.5 heads from this experiment. , > However, convergence issues associated with the infinite sum necessitate a more careful definition. is no longer guaranteed to be well defined at all. 's elements if summation is done row by row. The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes in a fair way between two players, who have to end their game before it is properly finished. {\displaystyle X^{+}(\omega )=\max(X(\omega ),0)} {\displaystyle \langle {\hat {A}}\rangle =\langle \psi |A|\psi \rangle } The expected value formula for a discrete random variable is: