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Mathematical Expectation (Expected Value) of a Random VariableIn probability theory the expected value (or mathematical expectation) of a random variable is the sum of the product of the values within the range of the discrete random variable and their respective probabilities of occurrence.If "x" is a value within the range of the discrete random variable "X" assuming the values x_{1}, x_{2}, x_{3}, ... x_{n}, with respective probabilities of occurrence p_{1}, p_{2}, p_{3}, ...., p_{n}, where p_{1} + p_{2} + p_{3} + ....+ p_{n} = 1, then the "Mathematical Expectation" or the "Expected Value" {represented by E(x)} of "x" is given by
Expectation of x^{r}The mathematical expectation of or the expected value of x^{r} is defined to be equal to the sum of the product of the values (within the range of the discrete random variable) raised to the power "r" and the probabilities of occurrence of the value.
Therefore, Expectation of x^{2} ⇒ E(x^{2}) = Σ p_{i} x_{i}^{2} Expectation of the sum of two or more variablesThe mathematical expectation of or the expected value of the sum of two or more variables is the sum of the expectations of the variables.E(x + y + ...) = E(x) + E(y) + ... Expectation of the product of two or more variablesThe mathematical expectation of or the expected value of the product of two or more independent variables is the product of the expectations of the variables.E(x × y × ...) = E(x) × E(y) × ... IllustrationIf "x" represents a value within the range of a random variable "X" and "c" is a constant show that (i) E(c) = c; (ii) E(cx) = c E(x) and (iii) E(x − ) = 0

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