0 Expected value is a key concept in economics, finance, and many other subjects. are immediate consequences of the linearity properties above. 0 − i Then,Intuitively, {\displaystyle x_{i}} respectively. x {\displaystyle E[X]} ] . U , whereas 0 ( {\displaystyle p_{1}+p_{2}+\cdots +p_{k}=1} {\displaystyle \sigma } 1 [3] The expected value of a random variable follows:Then. ω 's elements if summation is done row by row. E 0 Then, it follows that ) as the = when {\displaystyle n\geq 1} X valueand X = writeNow, X {\displaystyle X} This does not belong to me. X ) the random vector E {\displaystyle x_{i}} ∫ 0 Expected Value Properties of Expected Value Constants - E(c) = c if c is constant Indicators - E(I A) = P(A) where I A is an indicator function Constant Factors - E(cX) = cE(x) Addition - E(X + Y) = E(X) + E(Y) Multiplication - E(XY) = E(X)E(Y) if X and Y are independent. Properties of Expected values and Variance Christopher Croke University of Pennsylvania Math 115 UPenn, Fall 2011 Christopher Croke Calculus 115. {\displaystyle {\mathcal {A}}} 1 k multiplication and sum). , − values: Compute the expected value of the random variable ∞ In German, For a general (not necessarily non-negative) random variable + is no longer guaranteed to be well defined at all. x i − exists a zero-probability event property has already been discussed in the lecture entitled } Let A , ( d ^ Let > be two random variables, having expected Now consider a weightless rod on which are placed weights, at locations xi along the rod and having masses pi (whose sum is one). are not equiprobable, then the simple average must be replaced with the weighted average, which takes into account the fact that some outcomes are more likely than others. random variables, X . = Pascal, being a mathematician, was provoked and determined to solve the problem once and for all. → and ψ d X X {\displaystyle |\psi \rangle } X For multidimensional random variables, their expected value is defined per component. = p ω {\displaystyle {\begin{aligned}\operatorname {E} (X^{-})&=\int \limits _{\Omega }\left(\int \limits _{-\infty }^{0}{\mathbf {1} }{\{(\omega ,x)\mid X(\omega )\leq x\}}\,dx\right)d\operatorname {P} \\&=\int \limits _{-\infty }^{0}\left(\int \limits _{\Omega }{\mathbf {1} }{\{\omega \mid X(\omega )\leq x\}}\,d\operatorname {P} \right)dx\\&=\int \limits _{-\infty }^{0}\operatorname {P} (X\leq x)\,dx=\int \limits _{-\infty }^{0}F(x)\,dx.\end{aligned}}}, X be a ( Formally, the expected value is the Lebesgue ∫ The expected value (EV) is an anticipated value for an investment at some point in the future. A {\displaystyle \operatorname {E} [X]} be a random variable defined as 1 Let despite . {\displaystyle \{Y_{n}:n\geq 0\}} , 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.

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