In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the same experiment it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up is 3.5 as the number of rolls approaches infinity. In other words, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment.
More practically, the expected value of a discrete random variable is the probability-weighted average of all possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value. The same principle applies to an absolutely continuous random variable, except that an integral of the variable with respect to its probability density replaces the sum. The formal definition subsumes both of these and also works for distributions which are neither discrete nor absolutely continuous; the expected value of a random variable is the integral of the random variable with respect to its probability measure.
The expected value does not exist for random variables having some distributions with large "tails", such as the Cauchy distribution. For random variables such as these, the long-tails of the distribution prevent the sum or integral from converging.
The expected value is a key aspect of how one characterizes a probability distribution; it is one type of location parameter. By contrast, the variance is a measure of dispersion of the possible values of the random variable around the expected value. The variance itself is defined in terms of two expectations: it is the expected value of the squared deviation of the variable's value from the variable's expected value.
The expected value plays important roles in a variety of contexts. In regression analysis, one desires a formula in terms of observed data that will give a "good" estimate of the parameter giving the effect of some explanatory variable upon a dependent variable. The formula will give different estimates using different samples of data, so the estimate it gives is itself a random variable. A formula is typically considered good in this context if it is an unbiased estimator— that is if the expected value of the estimate (the average value it would give over an arbitrarily large number of separate samples) can be shown to equal the true value of the desired parameter.
In decision theory, and in particular in choice under uncertainty, an agent is described as making an optimal choice in the context of incomplete information. For risk neutral agents, the choice involves using the expected values of uncertain quantities, while for risk averse agents it involves maximizing the expected value of some objective function such as a von Neumann–Morgenstern utility function. One example of using expected value in reaching optimal decisions is the Gordon–Loeb model of information security investment. According to the model, one can conclude that the amount a firm spends to protect information should generally be only a small fraction of the expected loss (i.e., the expected value of the loss resulting from a cyber or information security breach).
- 1 Definition
- 2 Basic properties
- 2.2 If X = Y (a.s.) then E[X] = E[Y]
- 2.3 Expected value of a constant
- 2.4 Linearity
- 2.5 E[X] exists and is finite if and only if E[|X|] is finite
- 2.6 If X ≥ 0 (a.s.) then E[X] ≥ 0
- 2.7 Monotonicity
- 2.8 If (a.s.) and is finite then so is
- 2.9 If and then
- 2.10 Extremal property
- 2.11 Non-degeneracy
- 2.12 If then (a.s.)
- 2.14 Non-multiplicativity
- 2.15 Counterexample: despite pointwise
- 2.16 Countable non-additivity
- 2.17 Countable additivity for non-negative random variables
- 3 for independent and
- 4 Inequalities
- 5 Taking limits under the sign
- 6 Relationship with characteristic function
- 7 Uses and applications
- 8 The law of the unconscious statistician
- 9 Alternative formula for expected value
- 10 History
- 11 See also
- 12 Notes
- 13 Literature
Let be a random variable with a finite number of finite outcomes , , ..., occurring with probabilities , , ..., , respectively. The expectation of is defined as
Since all probabilities add up to 1 (), the expected value is the weighted average, with ’s being the weights.
If all outcomes are equiprobable (that is, ), then the weighted average turns into the simple average. This is intuitive: the expected value of a random variable is the average of all values it can take; thus the expected value is what one expects to happen on average. If the outcomes 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. The intuition however remains the same: the expected value of is what one expects to happen on average.
- Let represent the outcome of a roll of a fair six-sided die. More specifically, will be the number of pips showing on the top face of the die after the toss. The possible values for are 1, 2, 3, 4, 5, and 6, all equally likely (each having the probability of 1/). The expectation of is
- If one rolls the die times and computes the average (arithmetic mean) of the results, then as grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers. One example sequence of ten rolls of the die is 2, 3, 1, 2, 5, 6, 2, 2, 2, 6, which has the average of 3.1, with the distance of 0.4 from the expected value of 3.5. The convergence is relatively slow: the probability that the average falls within the range 3.5 ± 0.1 is 21.6% for ten rolls, 46.1% for a hundred rolls and 93.7% for a thousand rolls. See the figure for an illustration of the averages of longer sequences of rolls of the die and how they converge to the expected value of 3.5. More generally, the rate of convergence can be roughly quantified by e.g. Chebyshev's inequality and the Berry–Esseen theorem.
- 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 represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability 1/ in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be
- That is, the bet of $1 stands to lose $0.0526, so its expected value is -$0.0526.
Countably infinite case
Let be a random variable with a countable set of finite outcomes , , ..., occurring with probabilities , , ..., respectively, such that the infinite sum converges. The expected value of is defined as the series
Remark 1. Observe that
Remark 2. Due to absolute convergence, the expected value does not depend on the order in which the outcomes are presented. By contrast, a conditionally convergent series can be made to converge or diverge arbitrarily, via the Riemann rearrangement theorem.
- Suppose and for , where (with being the natural logarithm) is the scale factor such that the probabilities sum to 1. Then
- Since this series converges absolutely, the expected value of is .
- For an example that is not absolutely convergent, suppose random variable takes values 1, −2, 3, −4, ..., with respective probabilities , ..., where is a normalizing constant that ensures the probabilities sum up to one. Then the infinite sum
- converges and its sum is equal to . However it would be incorrect to claim that the expected value of is equal to this number—in fact does not exist (finite or infinite), as this series does not converge absolutely (see Alternating harmonic series).
- An example that diverges arises in the context of the St. Petersburg paradox. Let and for . The expected value calculation gives
- Since this does not converge but instead keeps growing, the expected value is infinite.
Absolutely continuous case
Remark. From computational perspective, the integral in the definition of may often be treated as an improper Riemann integral Specifically, if the function is Riemann-integrable on every finite interval , and
then the values (whether finite or infinite) of both integrals agree.
Remark 1. If and , then The functions and can be shown to be measurable (hence, random variables), and, by definition of Lebesgue integral,
where and are non-negative and possibly infinite.
The following scenarios are possible:
- is finite, i.e.
- is infinite, i.e. and
- is neither finite nor infinite, i.e.
Remark 2. If is the cumulative distribution function of , then
where the integral is interpreted in the sense of Lebesgue–Stieltjes.
Remark 3. An example of a distribution for which there is no expected value is Cauchy distribution.
Remark 4. For multidimensional random variables, their expected value is defined per component, i.e.
and, for a random matrix with elements ,
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The properties below replicate or follow immediately from those of Lebesgue integral.
If is an event, then where is the indicator function of the set .
Proof. By definition of Lebesgue integral of the simple function ,
If X = Y (a.s.) then E[X] = E[Y]
The statement follows from the definition of Lebesgue integral if we notice that (a.s.), (a.s.), and that changing a simple random variable on a set of probability zero does not alter the expected value.
Expected value of a constant
If is a random variable, and (a.s.), where , then . In particular, for an arbitrary random variable , .
Let be a constant random variable, i.e. . It follows from the definition of Lebesgue integral that . It also follows that (a.s.). By the previous property,
The expected value operator (or expectation operator) is linear in the sense that
where and are arbitrary random variables, and is a constant.
More rigorously, let and be random variables whose expected values are defined (different from ).
- If is also defined (i.e. differs from ), then
- Let be finite, and be a finite scalar. Then
1. We prove additivity in several steps.
1a. If and are simple and non-negative, taking intersections where necessary, one can re-write and in the form
for some measurable pairwise-disjoint sets partitioning , and being the indicator function of the set . By a straightforward check, the additivity follows.
1b. Assuming that and are arbitrary and non-negative, recall that every non-negative measurable function is a pointwise limit of a pointwise non-decreasing sequence of simple non-negative ones. Let and be such sequences converging to and respectively. We see that pointwise non-decreases, and pointwise. By monotone convergence theorem and case 1a,
(The reader can verify that using the monotone convergence theorem this way does not lead to circular logic).
1c. In the general case, if , then and
which is equivalent to,
2. To prove homogeneity, we first assume that the scalar above is non-negative. The finiteness of implies that is finite (a.s.). Therefore, is also finite (a.s.), which guarantees that is finite. The equality, thus, is a straightforward check based on the definition of Lebesgue integral.
If , then we first prove that by observing that and vice versa.
E[X] exists and is finite if and only if E[|X|] is finite
The following statements regarding a random variable are equivalent:
- exists and is finite.
- Both and are finite.
- is finite.
Sketch of proof. Indeed, . By linearity, . The above equivalency relies on the definition of Lebesgue integral and measurability of .
Remark. For the reasons above, the expressions " is integrable" and "the expected value of is finite" are used interchangeably when speaking of a random variable throughout this article.
If X ≥ 0 (a.s.) then E[X] ≥ 0
If , then , and hence, by definition of Lebesgue integral,
On the other hand, (a.s.), so, through a similar argument, , and therefore .
If (a.s.), and both and exist, then .
Remark. and exist in the sense that and
Proof follows from the linearity and the previous property if we set and notice that (a.s.).
If (a.s.) and is finite then so is
Let and be random variables such that (a.s.) and . Then .
Proof. Due to non-negativity of , exists, finite or infinite. By monotonicity, , so is finite which, as we saw earlier, is equivalent to being finite.
If and then
The proposition below will be used to prove the extremal property of later on.
Proposition. If is a random variable, then so is , for every . If, in addition, and , then .
|To see why the first statement holds, observe that is a composition of with . As a composition of two measurable functions, is measurable.
To prove the second statement, define
The reader can verify that is a random variable and . By non-negativity,
Counterexample for infinite measure
The requirement that is essential. By way of counterexample, consider the measurable space
where is the Borel -algebra on the interval and is the linear Lebesgue measure. The reader can prove that even though (Sketch of proof: and define a measure on Use "continuity from below" w.r. to and reduce to Riemann integral on each finite subinterval ).
Recall, as we proved early on, that if is a random variable, then so is .
Proposition (extremal property of ). Let be a random variable, and . Then and are finite, and is the best least squares approximation for among constants. Specifically,
- for every ,
- equality holds if and only if
( denotes the variance of ).
Remark (intuitive interpretation of extremal property). In intuitive terms, the extremal property says that if one is asked to predict the outcome of a trial of a random variable , then , in some practically useful sense, is one's best bet if no advance information about the outcome is available. If, on the other hand, one does have some advance knowledge regarding the outcome, then — again, in some practically useful sense — one's bet may be improved upon by using conditional expectations (of which is a special case) rather than .
Proof of proposition. By the above properties, both and are finite, and
whence the extremal property follows.
If , then (a.s.).
For every positive constant , . Indeed,
where is the indicator function of the set . By a property above, the finiteness of guarantees that the expected values and are also finite. By monotonicity,
For some integer , set . Define , and
The chain of sets
monotonically non-decreases, and . By "continuity from below", . Applying this formula, obtain
If then (a.s.)
Since is defined (i.e. ), and we know that is finite, and we want to show that (a.s.). We will show that where
If then and the proof is complete. Assuming that define
Given that , pick For every define
for some constant independent from (One can easily see that, in fact, but this is of no interest to us here).
Suppose that The sequence strictly increases, so, by definition of Lebesgue integral,
in contradiction with an earlier conclusion that is finite.
Corollary: if then (a.s.)
Corollary: if then (a.s.)
For an arbitrary random variable , .
Proof. By definition of Lebesgue integral,
Note that this result can also be proved based on Jensen's inequality.
In general, the expected value operator is not multiplicative, i.e. is not necessarily equal to . Indeed, let assume the values of 1 and -1 with probability 0.5 each. Then
The amount by which the multiplicativity fails is called the covariance:
If, however, the random variables and are independent, then , and .
Counterexample: despite pointwise
Let be the probability space, where is the Borel -algebra on and the linear Lebesgue measure. For define a sequence of random variables
and a random variable
on , with being the indicator function of the set .
For every as and
so On the other hand, and hence
In general, the expected value operator is not -additive, i.e.
By way of counterexample, let be the probability space, where is the Borel -algebra on and the linear Lebesgue measure. Define a sequence of random variables on , with being the indicator function of the set . For the pointwise sums, we have
By finite additivity,
On the other hand, and hence
Countable additivity for non-negative random variables
Let be non-negative random variables. It follows from monotone convergence theorem that
for independent and
Let and be independent random variables with finite expectations and . Then .
1. The case of non-negative -valued random variables.
Given a positive integer , let the random variables and assume their values in the set
Then , , and
where is the indicator function of the set ,
and denotes disjoint union. By definition of expected value,
Due to independence,
2. The case of non-negative random variables.
Let and be (arbitrary) non-negative random variable. Define
for an arbitrary . Note that is a random variable and
As we saw previously, the finiteness of implies that is finite almost sure, and consequently, (a.s.) on . This, in turn, implies that .
Let the random variable be defined the same way but with respect to . We have
and were shown to satisfy . Therefore,
It follows that, being independent from , the constant value can only be equal to 0.
3. The general case.
Let and be arbitrary random variables. We have
The Cauchy–Bunyakovsky–Schwarz inequality states that
For a nonnegative random variable and , the Markov's inequality states that
Let be an arbitrary random variable with finite expected value and finite variance . The Bienaymé-Chebyshev inequality states that, for any real number ,
Remark 1. The expected value is well-defined even if is allowed to assume infinite values. Indeed, implies that (a.s.), so the random variable is defined almost sure, and therefore there is enough information to compute
Remark 2. Jensen's inequality implies that since the absolute value function is convex.
Let . Lyapunov's inequality states that
Proof. Applying Jensen's inequality to and , obtain . Taking the th root of each side completes the proof.
Let and satisfy , , and . The Hölder's inequality states that
Let be an integer satisfying . Let, in addition, and . Then, according to the Minkowski inequality, and
Taking limits under the sign
Monotone convergence theorem
Let the sequence of random variables and the random variables and be defined on the same probability space Suppose that
- all the expected values and are defined (differ from );
- for every
- is the pointwise limit of (a.s.), i.e. (a.s.).
The monotone convergence theorem states that
Observe that, by monotonicity, the sequence monotonically non-decreases, and
If then and we are done.
If then, following the assumption that we conclude that is finite which, in turn, implies, as we saw previously, that is finite (a.s.).
Denote and . The finiteness of (a.s.) implies that the differences and are defined (do not have the form ) everywhere outside of a null set. On that null set, and may be defined arbitrarily (e.g. as zero or in any other way, as long as measurability is preserved) without affecting this proof. As a difference of two random variables, and are also random variables.
It follows from the definition that (a.s.), (a.s.), the sequence pointwise non-decreases (a.s.), and pointwise (a.s.).
By (the general version of) monotone convergence theorem,
whence the assertion follows.
Let the sequence of random variables and the random variable be defined on the same probability space Suppose that
- all the expected values and are defined (differ from );
- (a.s.), for every
Fatou's lemma states that
(Note that is a random variable, for every by the properties of limit inferior).
If then, by monotonicity, so and the assertion follows.
If , then, following the assumption that we conclude that is finite which, in turn, implies, as we saw previously, that is finite (a.s.).
Denote . Then (a.s.). The finiteness of (a.s.) implies that is defined (does not have the form ) everywhere outside of a null set. On that null set may be defined arbitrarily (e.g. as zero or in any other way, as long as measurability is preserved) without affecting this proof. As a difference of two random variables, is a random variable.
By (the general version of) Fatou's lemma,
whence the assertion follows.
- pointwise (a.s.);
- for some constant (independent from );
- (a.s.), for every
Proof is by observing that (a.s.) and applying Fatou's lemma.
Dominated convergence theorem
Let be a sequence of random variables. If pointwise (a.s.), (a.s.), and . Then, according to the dominated convergence theorem,
- the function is measurable (hence a random variable);
- all the expected values and are defined (do not have the form );
- (both sides may be infinite);
Relationship with characteristic function
The probability density function of a scalar random variable is related to its characteristic function by the inversion formula:
For the expected value of (where is a Borel function), we can use this inversion formula to obtain
If is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem,
is the Fourier transform of The expression for also follows directly from Plancherel theorem.
Uses and applications
It is possible to construct an expected value equal to the probability of an event by taking the expectation of an indicator function that is one if the event has occurred and zero otherwise. This relationship can be used to translate properties of expected values into properties of probabilities, e.g. using the law of large numbers to justify estimating probabilities by frequencies.
The expected values of the powers of X are called the moments of X; the moments about the mean of X are expected values of powers of X − E[X]. The moments of some random variables can be used to specify their distributions, via their moment generating functions.
To empirically estimate the expected value of a random variable, one repeatedly measures observations of the variable and computes the arithmetic mean of the results. If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals (the sum of the squared differences between the observations and the estimate). The law of large numbers demonstrates (under fairly mild conditions) that, as the size of the sample gets larger, the variance of this estimate gets smaller.
This property is often exploited in a wide variety of applications, including general problems of statistical estimation and machine learning, to estimate (probabilistic) quantities of interest via Monte Carlo methods, since most quantities of interest can be written in terms of expectation, e.g. , where is the indicator function of the set .
In classical mechanics, the center of mass is an analogous concept to expectation. For example, suppose X is a discrete random variable with values xi and corresponding probabilities pi. Now consider a weightless rod on which are placed weights, at locations xi along the rod and having masses pi (whose sum is one). The point at which the rod balances is E[X].
A very important application of the expectation value is in the field of quantum mechanics. The expectation value of a quantum mechanical operator operating on a quantum state vector is written as . The uncertainty in can be calculated using the formula .
The law of the unconscious statistician
The expected value of a measurable function of , , given that has a probability density function , is given by the inner product of and :
Alternative formula for expected value
Formula for non-negative random variables
Finite and countably infinite case
For a non-negative integer-valued random variable
If then On the other hand,
so the series on the right diverges to and the equality holds.
be an infinite upper triangular matrix. The double series is the sum of 's elements if summation is done row by row. Since every summand is non-negative, the series either converges absolutely or diverges to In both cases, changing summation order does not affect the sum. Changing summation order, from row-by-row to column-by-column, gives us
In a coin tossing experiment, let the probability of heads be . Including the final attempt, how many tosses can we expect until the first head?
Solution. If is the random variable indicating the numbers of coin tosses before and including the first head, then, for ,
where we took into account the geometric series summation formula. We now compute
If is a non-negative random variable, then
where denotes improper Riemann integral.
1. For every ,
where and are the indicator functions of and , respectively. Substituting this into the definition of , obtain
Since and this integral (finite or infinite) meets the requirements of Tonelli's theorem. Changing the order of integration gives us
2a. The function is Riemann-integrable on each finite interval Indeed, since is non-increasing, the set of its discontinuities is countable. Due to countable additivity, is a null set with respect to the linear Lebesgue measure. Furthermore, for all Using the Lebesgue criterion, Riemann integrability of follows. We also conclude that
2b. By "continuity from below",
The case of is similar.
Formula for non-positive random variables
If is a non-positive random variable, then
where denotes improper Riemann integral.
This formula follows from that for the non-negative case applied to
If, in addition, is integer-valued, i.e. , then
If can be both positive and negative, then , and the above results may be applied to and separately.
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. 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.
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." (cited by 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.
Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes: "That my Chance or Expectation to win any thing is worth just such a Sum, as wou'd procure me 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/." 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:
… 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.
The use of the letter E to denote expected value goes back to W.A. Whitworth in 1901, who used a script E. The symbol has become popular since for English writers it meant "Expectation", for Germans "Erwartungswert", for Spanish "Esperanza matemática" and for French "Espérance mathématique".
- Center of mass
- Central tendency
- Chebyshev's inequality (an inequality on location and scale parameters)
- Conditional expectation
- Expected value is also a key concept in economics, finance, and many other subjects
- The general term expectation
- Expectation value (quantum mechanics)
- Law of total expectation –the expected value of the conditional expected value of X given Y is the same as the expected value of X.
- Moment (mathematics)
- Nonlinear expectation (a generalization of the expected value)
- Wald's equation for calculating the expected value of a random number of random variables
- Sheldon M Ross (2007). "§2.4 Expectation of a random variable". Introduction to probability models (9th ed.). Academic Press. p. 38 ff. ISBN 0-12-598062-0.
- Richard W Hamming (1991). "§2.5 Random variables, mean and the expected value". The art of probability for scientists and engineers. Addison–Wesley. p. 64 ff. ISBN 0-201-40686-1.
- Richard W Hamming (1991). "Example 8.7–1 The Cauchy distribution". The art of probability for scientists and engineers. Addison-Wesley. p. 290 ff. ISBN 0-201-40686-1.
Sampling from the Cauchy distribution and averaging gets you nowhere — one sample has the same distribution as the average of 1000 samples!
- Gordon, Lawrence; Loeb, Martin (November 2002). "The Economics of Information Security Investment". ACM Transactions on Information and System Security. 5 (4): 438–457. doi:10.1145/581271.581274.
- Expectation Value, retrieved August 8, 2017
- Papoulis, A. (1984), Probability, Random Variables, and Stochastic Processes, New York: McGraw–Hill, pp. 139–152
- "Ore, Pascal and the Invention of Probability Theory". The American Mathematical Monthly. 67 (5): 409–419. 1960. doi:10.2307/2309286.
- Whitworth, W.A. (1901) Choice and Chance with One Thousand Exercises. Fifth edition. Deighton Bell, Cambridge. [Reprinted by Hafner Publishing Co., New York, 1959.]
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