Expected Value

In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable.

The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration.

The expected value of a random variable is often denoted by , , or , with also often stylized as or $\mathbb.$

History

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. This problem had been debated for centuries. Many conflicting proposals and solutions had been suggested over the years when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem couldn't be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all.

He began to discuss the problem in the famous series of letters to Pierre de Fermat. Soon enough, they both independently came up with a solution. They solved the problem in different computational ways, but their results were identical because their computations were based on the same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively; however, they did not publish their findings. They only informed a small circle of mutual scientific friends in Paris about it.

In Dutch mathematician Christiaan Huygens' book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens published his treatise in 1657, (see Huygens (1657)) "''De ratiociniis in ludo aleæ''" on probability theory just after visiting Paris. The book extended the concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players), and can be seen as the first successful attempt at laying down the foundations of the theory of probability.

In the foreword to his treatise, Huygens wrote:

During his visit to France in 1655, Huygens learned about de Méré's Problem. From his correspondence with Carcavine a year later (in 1656), he realized his method was essentially the same as Pascal's. Therefore, he knew about Pascal's priority in this subject before his book went to press in 1657.

In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the expectations of random variables.

Etymology

Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes:

More than a hundred years later, in 1814, Pierre-Simon Laplace published his tract "''Théorie analytique des probabilités''", where the concept of expected value was defined explicitly:

Notations

The use of the letter to denote expected value goes back to W. A. Whitworth in 1901. The symbol has become popular since then for English writers. In German, stands for "Erwartungswert", in Spanish for "Esperanza matemática", and in French for "Espérance mathématique".

When "E" is used to denote expected value, authors use a variety of stylization: the expectation operator can be stylized as (upright), (italic), or $\mathbb$ (in blackboard bold), while a variety of bracket notations (such as , , and ) are all used.

Another popular notation is , whereas , , and $\overline$ are commonly used in physics, and in Russian-language literature.

Definition

As discussed below, there are several context-dependent ways of defining the expected value. The simplest and original definition deals with the case of finitely many possible outcomes, such as in the flip of a coin. With the theory of infinite series, this can be extended to the case of countably many possible outcomes. It is also very common to consider the distinct case of random variables dictated by (piecewise-)continuous probability density functions, as these arise in many natural contexts. All of these specific definitions may be viewed as special cases of the general definition based upon the mathematical tools of measure theory and Lebesgue integration, which provide these different contexts with an axiomatic foundation and common language.

Any definition of expected value may be extended to define an expected value of a multidimensional random variable, i.e. a random vector . It is defined component by component, as . Similarly, one may define the expected value of a random matrix with components by .

Random variables with finitely many outcomes

Consider a random variable with a ''finite'' list of possible outcomes, each of which (respectively) has probability of occurring. The expectation of is defined as

:\operatorname =x_1p_1 + x_2p_2 + \cdots + x_kp_k.

Since the probabilities must satisfy , it is natural to interpret as a weighted average of the values, with weights given by their probabilities .

In the special case that all possible outcomes are equiprobable (that is, ), the weighted average is given by the standard average. In the general case, the expected value takes into account the fact that some outcomes are more likely than others.

Examples

*Let $X$ represent the outcome of a roll of a fair six-sided . More specifically, $X$ will be the number of pips showing on the top face of the after the toss. The possible values for $X$ are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of . The expectation of $X$ is
:: \operatorname = 1\cdot\frac16 + 2\cdot\frac16 + 3\cdot\frac16 + 4\cdot\frac16 + 5\cdot\frac16 + 6\cdot\frac16 = 3.5.

:If one rolls the $n$ times and computes the average (arithmetic mean) of the results, then as $n$ grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers.

*The roulette game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable $X$ represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be
:: \operatorname ,\text\$1\text\,= -\$1 \cdot \frac + \$35 \cdot \frac = -\$\frac.

:That is, the expected value to be won from a $1 bet is −$. Thus, in 190 bets, the net loss will probably be about $10.

Random variables with countably many outcomes

Informally, the expectation of a random variable with a countable set of possible outcomes is defined analogously as the weighted average of all possible outcomes, where the weights are given by the probabilities of realizing each given value. This is to say that
: \operatorname = \sum_^\infty x_i\, p_i,
where are the possible outcomes of the random variable and are their corresponding probabilities. In many non-mathematical textbooks, this is presented as the full definition of expected values in this context.

However, there are some subtleties with infinite summation, so the above formula is not suitable as a mathematical definition. In particular, the Riemann series theorem of mathematical analysis illustrates that the value of certain infinite sums involving positive and negative summands depends on the order in which the summands are given. Since the outcomes of a random variable have no naturally given order, this creates a difficulty in defining expected value precisely.

For this reason, many mathematical textbooks only consider the case that the infinite sum given above converges absolutely, which implies that the infinite sum is a finite number independent of the ordering of summands. In the alternative case that the infinite sum does not converge absolutely, one says the random variable ''does not have finite expectation.''

Examples

*Suppose $x_i = i$ and $p_i = \tfrac$ for $i = 1, 2, 3, \ldots,$ where $c = \tfrac$ is the scaling factor which makes the probabilities sum to 1. Then, using the direct definition for non-negative random variables, we have \operatorname \,= \sum_i x_i p_i = 1(\tfrac)
+ 2(\tfrac) + 3 (\tfrac) + \cdots
\,= \, \tfrac + \tfrac + \tfrac + \cdots \,=\, c \,=\, \tfrac.

Random variables with density

Now consider a random variable which has a probability density function given by a function on the real number line. This means that the probability of taking on a value in any given open interval is given by the integral of over that interval. The expectation of is then given by the integral
: \operatorname = \int_^\infty x f(x)\, dx.
A general and mathematically precise formulation of this definition uses measure theory and Lebesgue integration, and the corresponding theory of ''absolutely continuous random variables'' is described in the next section. The density functions of many common distributions are piecewise continuous, and as such the theory is often developed in this restricted setting. For such functions, it is sufficient to only consider the standard Riemann integration. Sometimes ''continuous random variables'' are defined as those corresponding to this special class of densities, although the term is used differently by various authors.

Analogously to the countably-infinite case above, there are subtleties with this expression due to the infinite region of integration. Such subtleties can be seen concretely if the distribution of is given by the Cauchy distribution , so that . It is straightforward to compute in this case that
:$\int_a^b xf(x)\,dx=\int_a^b \frac\,dx=\frac\ln\frac.$
The limit of this expression as and does not exist: if the limits are taken so that , then the limit is zero, while if the constraint is taken, then the limit is .

To avoid such ambiguities, in mathematical textbooks it is common to require that the given integral converges absolutely, with left undefined otherwise. However, measure-theoretic notions as given below can be used to give a systematic definition of for more general random variables .

Arbitrary real-valued random variables

All definitions of the expected value may be expressed in the language of measure theory. In general, if is a real-valued random variable defined on a probability space , then the expected value of , denoted by , is defined as the Lebesgue integral
:\operatorname = \int_\Omega X\,d\operatorname.
Despite the newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the Lebesgue integral of is defined via weighted averages of ''approximations'' of which take on finitely many values. Moreover, if given a random variable with finitely or countably many possible values, the Lebesgue theory of expectation is identical with the summation formulas given above. However, the Lebesgue theory clarifies the scope of the theory of probability density functions. A random variable is said to be ''absolutely continuous'' if any of the following conditions are satisfied:
* there is a nonnegative measurable function on the real line such that
::$\text(X\in A)=\int_A f(x)\,dx,$
:for any Borel set , in which the integral is Lebesgue.
* the cumulative distribution function of is absolutely continuous.
* for any Borel set of real numbers with Lebesgue measure equal to zero, the probability of being valued in is also equal to zero
* for any positive number there is a positive number such that: if is a Borel set with Lebesgue measure less than , then the probability of being valued in is less than .
These conditions are all equivalent, although this is nontrivial to establish. In this definition, is called the ''probability density function'' of (relative to Lebesgue measure). According to the change-of-variables formula for Lebesgue integration, combined with the law of the unconscious statistician, it follows that
:\operatorname equiv\int_\Omega X\,d\operatorname=\int_xf(x)\,dx
for any absolutely continuous random variable . The above discussion of continuous random variables is thus a special case of the general Lebesgue theory, due to the fact that every piecewise-continuous function is measurable.

Infinite expected values

Expected values as defined above are automatically finite numbers. However, in many cases it is fundamental to be able to consider expected values of . This is intuitive, for example, in the case of the St. Petersburg paradox, in which one considers a random variable with possible outcomes , with associated probabilities , for ranging over all positive integers. According to the summation formula in the case of random variables with countably many outcomes, one has
\operatorname \sum_^\infty x_i\,p_i =2\cdot \frac+4\cdot\frac + 8\cdot\frac+ 16\cdot\frac+ \cdots = 1 + 1 + 1 + 1 + \cdots.
It is natural to say that the expected value equals .

There is a rigorous mathematical theory underlying such ideas, which is often taken as part of the definition of the Lebesgue integral. The first fundamental observation is that, whichever of the above definitions are followed, any ''nonnegative'' random variable whatsoever can be given an unambiguous expected value; whenever absolute convergence fails, then the expected value can be defined as . The second fundamental observation is that any random variable can be written as the difference of two nonnegative random variables. Given a random variable , one defines the positive and negative parts by and . These are nonnegative random variables, and it can be directly checked that . Since and are both then defined as either nonnegative numbers or , it is then natural to define:

\operatorname = \begin \operatorname ^+- \operatorname ^-& \text \operatorname ^+< \infty \text \operatorname ^-< \infty;\\
+\infty & \text \operatorname ^+= \infty \text \operatorname ^-< \infty;\\
-\infty & \text \operatorname ^+< \infty \text \operatorname ^-= \infty;\\
\text & \text \operatorname ^+= \infty \text \operatorname ^-= \infty.
\end

According to this definition, exists and is finite if and only if and are both finite. Due to the formula , this is the case if and only if is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite expected values in any cases not previously considered; they are only useful for infinite expectations.
*In the case of the St. Petersburg paradox, one has and so as desired.
* Suppose the random variable takes values with respective probabilities . Then it follows that takes value with probability for each positive integer , and takes value with remaining probability. Similarly, takes value with probability for each positive integer and takes value with remaining probability. Using the definition for non-negative random variables, one can show that both and (see Harmonic series). Hence, in this case the expectation of is undefined.
* Similarly, the Cauchy distribution, as discussed above, has undefined expectation.

Expected values of common distributions

The following table gives the expected values of some commonly occurring probability distributions. The third column gives the expected values both in the form immediately given by the definition, as well as in the simplified form obtained by computation therefrom. The details of these computations, which are not always straightforward, can be found in the indicated references.

Properties

The basic properties below (and their names in bold) replicate or follow immediately from those of Lebesgue integral. Note that the letters "a.s." stand for "almost surely"—a central property of the Lebesgue integral. Basically, one says that an inequality like $X \geq 0$ is true almost surely, when the probability measure attributes zero-mass to the complementary event $\left\$.

*Non-negativity: If $X \geq 0$ (a.s.), then \operatorname X\geq 0.
*Linearity of expectation: The expected value operator (or expectation operator) $\operatorname$cdot/math> is linear in the sense that, for any random variables $X$ and $Y$, and a constant $a$, \begin
\operatorname + Y&= \operatorname + \operatorname \\
\operatorname X &= a \operatorname
\end

:whenever the right-hand side is well-defined. By induction, this means that the expected value of the sum of any finite number of random variables is the sum of the expected values of the individual random variables, and the expected value scales linearly with a multiplicative constant. Symbolically, for $N$ random variables $X_$ and constants $a_ (1\leq i \leq N)$, we have \operatorname\left sum_^a_X_\right= \sum_^a_\operatorname _. If we think of the set of random variables with finite expected value as forming a vector space, then the linearity of expectation implies that the expected value is a linear form on this vector space.
*Monotonicity: If $X\leq Y$ (a.s.), and both \operatorname /math> and \operatorname /math> exist, then \operatorname leq\operatorname /math>. Proof follows from the linearity and the non-negativity property for $Z=Y-X$, since $Z\geq 0$ (a.s.).
*Non-degeneracy: If \operatorname (a.s.),_then__\operatorname_X=_\operatorname Y/math>._In_other_words,_if_X_and_Y_are_random_variables_that_take_different_values_with_probability_zero,_then_the_expectation_of_X_will_equal_the_expectation_of_Y.
*_If_$X=c$__(a.s.)_for_some_real_number_,_then_\operatorname_=_c._In_particular,_for_a_random_variable_$X$_with_well-defined_expectation,_\operatorname operatorname[X_=_\operatorname_/math>._A_well_defined_expectation_implies_that_there_is_one_number,_or_rather,_one_constant_that_defines_the_expected_value._Thus_follows_that_the_expectation_of_this_constant_is_just_the_original_expected_value.
*_As_a_consequence_of_the_formula__as_discussed_above,_together_with_the_triangle_inequality,_it_follows_that_for_any_random_variable_$X$_with_well-defined_expectation,_one_has__, \operatorname _\leq_\operatorname, X, _.
*Let__denote_the_ indicator_function_of_an_event_,_then__is_given_by_the_probability_of_._This_is_nothing_but_a_different_way_of_stating_the_expectation_of_a_Bernoulli_random_variable,_as_calculated_in_the_table_above.
*

Formulas_in_terms_of_CDF:_If_$F(x)$_is_the_cumulative_distribution_function_

In_probability_theory_and__statistics,_the_cumulative_distribution_function_(CDF)_of_a_real-valued__random_variable_X,_or_just_distribution_function_of_X,_evaluated_at_x,_is_the_probability_that_X_will_take_a_value_less_than_or_equal_to_x.

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