Expected Value

For example, an American Roulette wheel has 38 equally possible outcomes. A 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 the expected value of the profit resulting from a $1 bet on a single number is, considering all 38 possible outcomes: : $\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. MATHEMATICAL DEFINITION In general, if $X\,$ is a Random Variable defined on a Probability Space $(\Omega, P)\,$ , then the expected value of $X\,$ (denoted $\mathrm{E}(X)\,$ or sometimes $\langle X angle$ or $\mathbb{E}(X)$ ) is defined as : $\mathrm{E}(X) = \int_\Omega X\, dP$ 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. If $X$ is a Discrete Random Variable with values $x_1$ , $x_2$ , ... and corresponding probabilities $p_1$ , $p_2$ , ... which add up to 1, then $\mathrm{E}(X)$ can be computed as the sum or Series : $\mathrm{E}(X) = \sum_i p_i x_i\,$ 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 : $\mathrm{E}(X) = \int_{-\infty}^\infty x f(x)\, \mathrm d x.$ 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 : $\mathrm{E}(g(X)) = \int_{-\infty}^\infty g(x) f(x)\, \mathrm d x.$ 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 means the expected value of a random variable that is the number of such attempts. And so on. PROPERTIES Linearity The expected value operator (or expectation operator) $\mathrm{E}$ is Linear in the sense that : $\mathrm{E}(a X + b Y) = a \mathrm{E}(X) + b \mathrm{E}(Y)\,$ 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 For any two random variables $X,Y$ one may define the Conditional Expectation :
	\mathrm{E} \left( \mathrm{E}[XY] Ight) &	& \sum_y \mathrm{E}[XY=y] \cdot \mathrm{P}(Y=y) \
	&	& \sum_y \left( \sum_x x \cdot \mathrm{P}(X=xY=y) ight) \cdot \mathrm{P}(Y=y) \
	&	& \sum_y \sum_x x \cdot \mathrm{P}(X=xY=y) \cdot \mathrm{P}(Y=y) \
	&	& \sum_y \sum_x x \cdot \mathrm{P}(Y=yX=x) \cdot \mathrm{P}(X=x) \
	&	& \sum_x x \cdot \mathrm{P}(X=x) \cdot \left( \sum_y \mathrm{P}(Y=yX=x) ight) \
	:<math>\mathrm{E}	"X]" class="copylinks" target="_blank">= \mathrm{E} \left( \mathrm{E}[XY ight)</math>
	:<math>\mathrm{E}	"X]" class="copylinks" target="_blank">\leq \mathrm{E}[X </math>

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Expected Value