Expected value

In probability theory, the expected value of a random variable is closely related to the weighted average and intuitively is the arithmetic mean of a large number of independent realizations of that variable. The expected value is also known as the expectation, mathematical expectation, mean, average, or first moment.

By definition, the expected value of a constant random variable ${\displaystyle X=c}$ is ${\displaystyle c}$. The expected value of a random variable ${\displaystyle X}$ with equiprobable outcomes ${\displaystyle \{c_{1},\ldots ,c_{n}\}}$ is defined as the arithmetic mean of the terms ${\displaystyle c_{i}.}$ If some of the probabilities ${\displaystyle \Pr \,(X=c_{i})}$ of an individual outcome ${\displaystyle c_{i}}$ are unequal, then the expected value is defined to be the probability-weighted average of the ${\displaystyle c_{i}}$s, i.e. the sum of the ${\displaystyle n}$ products ${\displaystyle c_{i}\cdot \Pr \,(X=c_{i})}$.

Expected value of a general random variable is defined in a way that extends the notion of probability-weighted average and involves integration in the sense of Lebesgue.

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's properly finished.[1] This problem had been debated for centuries, and many conflicting proposals and solutions had been suggested over the years, when it was posed in 1654 to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré. 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 a now 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 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.[2]

Three years later, in 1657, a Dutch mathematician Christiaan Huygens, who had just visited Paris, published a treatise (see Huygens (1657)) "De ratiociniis in ludo aleæ" on probability theory. In this book he considered the problem of points and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens also 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). In this sense this book can be seen as the first successful attempt at laying down the foundations of the theory of probability.

In the foreword to his book, Huygens wrote:

It should be said, also, that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me the honour of the first invention. This does not belong to me. But these savants, although they put each other to the test by proposing to each other many questions difficult to solve, have hidden their methods. I have had therefore to examine and go deeply for myself into this matter by beginning with the elements, and it is impossible for me for this reason to affirm that I have even started from the same principle. But finally I have found that my answers in many cases do not differ from theirs.

— Edwards (2002)

Thus, Huygens learned about de Méré's Problem in 1655 during his visit to France; later on in 1656 from his correspondence with Carcavi he learned that his method was essentially the same as Pascal's; so that before his book went to press in 1657 he knew about Pascal's priority in this subject.

Etymology

Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes:[3]

That any one Chance or Expectation to win any thing is worth just such a Sum, as wou'd procure in the same Chance and Expectation at a fair Lay. ... If I expect a or b, and have an equal chance of gaining them, my Expectation is worth (a+b)/2.

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:[4]

… this advantage in the theory of chance is the product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right for the sum hoped for. We will call this advantage mathematical hope.

Notations

The use of the letter ${\displaystyle \mathop {\hbox{E}} }$  to denote expected value goes back to W. A. Whitworth in 1901.[5] The symbol has become popular since for English writers. In German, ${\displaystyle \mathop {\hbox{E}} }$  stands for "Erwartungswert", in Spanish for "Esperanza matemática", and in French for "Espérance mathématique".[6]

Another popular notation is ${\displaystyle \mu _{X}}$ , where as ${\displaystyle \langle X\rangle }$  is commonly used in physics, and ${\displaystyle \mathop {\hbox{M}} (X)}$  in Russian-language literature.

Definition

Finite case

Let ${\displaystyle X}$  be a random variable with a finite number of finite outcomes ${\displaystyle x_{1},x_{2},\ldots ,x_{k}}$  occurring with probabilities ${\displaystyle p_{1},p_{2},\ldots ,p_{k},}$  respectively. The expectation of ${\displaystyle X}$  is defined as

${\displaystyle \operatorname {E} [X]=\sum _{i=1}^{k}x_{i}\,p_{i}=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}.}$

Since the sum of all probabilities ${\displaystyle p_{i}}$  is 1 (${\displaystyle p_{1}+p_{2}+\cdots +p_{k}=1}$ ), the expected value is the weighted average of the ${\displaystyle x_{i}}$  values, with the ${\displaystyle p_{i}}$  values being the weights.

If all outcomes ${\displaystyle x_{i}}$  are equiprobable (that is, ${\displaystyle p_{1}=p_{2}=\cdots =p_{k}}$ ), then the weighted average turns into the simple average. If the outcomes ${\displaystyle x_{i}}$  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 the others.

An illustration of the convergence of sequence averages of rolls of a die to the expected value of 3.5 as the number of rolls (trials) grows.

Examples

• Let ${\displaystyle X}$  represent the outcome of a roll of a fair six-sided die. More specifically, ${\displaystyle X}$  will be the number of pips showing on the top face of the die after the toss. The possible values for ${\displaystyle X}$  are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of 1/6. The expectation of ${\displaystyle X}$  is
${\displaystyle \operatorname {E} [X]=1\cdot {\frac {1}{6}}+2\cdot {\frac {1}{6}}+3\cdot {\frac {1}{6}}+4\cdot {\frac {1}{6}}+5\cdot {\frac {1}{6}}+6\cdot {\frac {1}{6}}=3.5.}$
If one rolls the die ${\displaystyle n}$  times and computes the average (arithmetic mean) of the results, then as ${\displaystyle n}$  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 ${\displaystyle X}$  represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability 1/38 in American roulette), the payoff is$35; otherwise the player loses the bet. The expected profit from such a bet will be
${\displaystyle \operatorname {E} [\,{\text{gain from }}\1{\text{ bet}}\,]=-\1\cdot {\frac {37}{38}}+\35\cdot {\frac {1}{38}}=-\{\frac {1}{19}}.}$

Literature

• Edwards, A.W.F (2002). Pascal's arithmetical triangle: the story of a mathematical idea (2nd ed.). JHU Press. ISBN 0-8018-6946-3.CS1 maint: ref=harv (link)
• Huygens, Christiaan (1657). De ratiociniis in ludo aleæ (English translation, published in 1714).CS1 maint: ref=harv (link)