Expected Value

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Expected Value

For example, the expected value from the roll of an ordinary six-sided Die is 3.5, found by, : $\begin{align} \operatorname{E}(X)& = 1 \cdot rac{1}{6} + 2 \cdot rac{1}{6} + 3 \cdot rac{1}{6} + 4 \cdot rac{1}{6} + 5 \cdot rac{1}{6} + 6 \cdot rac{1}{6}\$ {Link without Title} & = rac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5, \end{align} which is not one of the possible outcomes. A common application of expected value is in Gambling . For example, an American Roulette wheel has 38 equally likely outcomes. A winning bet placed on a single number pays 35-to-1 (this means that you are paid 35 times your bet and your bet is returned, so you get 36 times your bet). So considering all 38 possible outcomes, the expected value of the profit resulting from a $1 bet on a single number is: : $\left( -$1 imes rac{37}{38} ight) + \left( $35 imes rac{1}{38} ight),$ which is about −$0.0526. Therefore one expects, on average, to lose over five cents for every dollar bet, and the expected value of a one dollar bet is $0.9474. In gambling or betting, a game or situation in which the expected value of the profit for the player is zero (no net gain nor loss) is commonly called a "fair game." MATHEMATICAL DEFINITION In general, if $X\,$ is a Random Variable defined on a Probability Space $(\Omega, \Sigma, P)\,$ , then the expected value of $X\,$ (denoted $\operatorname{E}(X)\,$ or sometimes $\langle X angle$ or $\mathbb{E}(X)$ ) is defined as : $\operatorname{E}(X) = \int_\Omega X\, \operatorname{d}P$ where the Lebesgue Integral is employed. Note that not all random variables have an expected value, since the integral may not exist (e.g., Cauchy Distribution ). Two variables with the same Probability Distribution will have the same expected value, if it is defined. : $\operatorname{E}(X) = \sum_i p_i x_i\,$ as in the gambling example mentioned above. If the Probability Distribution of $X$ admits a Probability Density Function $f(x)$ , then the expected value can be computed as : $\operatorname{E}(X) = \int_{-\infty}^\infty x f(x)\, \operatorname{d}x .$ It follows directly from the discrete case definition that if $X$ is a Constant Random Variable , i.e. $X = b$ for some fixed Real Number $b$ , then the expected value of $X$ is also $b$ . The expected value of an arbitrary function of ''X'', ''g(X)'', with respect to the probability density function ''f(x)'' is given by: : $\operatorname{E}(g(X)) = \int_{-\infty}^\infty g(x) f(x)\, \operatorname{d}x .$ CONVENTIONAL TERMINOLOGY When one speaks of the "expected price", one means the expected value of a random variable that is a price. When one speaks of the "expected height", one means the expected value of a random variable that is a height. When one speaks of the "expected number of attempts needed to get one successful attempt," one might conservatively approximate it as the reciprocal of the probability of success for such an attempt. And so on. PROPERTIES Constants Expected value of a constant is equal to that constant or If c is a constant, E(c) = c Monotonicity If ''X'' and ''Y'' are random variables so that $X \le Y$ Almost Surely , then $\operatorname{E}(X) \le \operatorname{E}(Y)$ . Linearity The expected value operator (or expectation operator) $\operatorname{E}$ is Linear in the sense that : $\operatorname{E}(X + c)= \operatorname{E}(X) + c\,$ : $\operatorname{E}(X + Y)= \operatorname{E}(X) + \operatorname{E}(Y)\,$ : $\operatorname{E}(aX)= a \operatorname{E}(X)\,$ Combining the results from previous three equations, we can see that - : $\operatorname{E}(aX + b)= a \operatorname{E}(X) + b\,$ : $\operatorname{E}(a X + b Y) = a \operatorname{E}(X) + b \operatorname{E}(Y)\,$ for any two random variables $X$ and $Y$ (which need to be defined on the same probability space) and any real numbers $a$ and $b$ . Iterated expectation Iterated expectation for discrete random variables For any two Discrete random variables $X,Y$ one may define the Conditional Expectation :
	\operatorname{E} \left( \operatorname{E}(XY) Ight)	\sum\limits_y \operatorname{E}(XY=y) \cdot \operatorname{P}(Y=y) \,</math>
	:::<math>	\sum\limits_y \left( \sum\limits_x x \cdot \operatorname{P}(X=xY=y) ight) \cdot \operatorname{P}(Y=y)\, </math>
	:::<math>	\sum\limits_y \sum\limits_x x \cdot \operatorname{P}(X=xY=y) \cdot \operatorname{P}(Y=y)\, </math>
	:::<math>	\sum\limits_y \sum\limits_x x \cdot \operatorname{P}(Y=yX=x) \cdot \operatorname{P}(X=x) \, </math>
	:::<math>	\sum\limits_x x \cdot \operatorname{P}(X=x) \cdot \left( \sum\limits_y \operatorname{P}(Y=yX=x) ight) \, </math>
	:<math>\operatorname{E}(X)	\operatorname{E} \left( \operatorname{E}(XY) ight)</math>
	:<math>\operatorname{E}(X)	\operatorname{E} \left( \operatorname{E}(XY) ight)</math>