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Expected value

From Wikipedia, the free encyclopedia

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 (see § Examples for details). 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.[1][2]

The expected value does not exist for random variables having some distributions with large "tails", such as the Cauchy distribution.[3] 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 (var(X) = E(X2) - [E(X)]2).

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).[4]

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Transcription

When we first started talking about central tendencies and how we measure average, we talked about the arithmetic mean and there you just added up the numbers and you divided by the number of numbers there were. So let's say our population of numbers is-- we have a 3. Let's say we have three 3's, a 4, and a 5. That's our population. And if we wanted the population mean here we were just add all the numbers up. We'd say 3 plus 3 plus 3 plus 4 plus 5. And then we would divide that sum by the number of numbers we have. We would divide that by 5. And we would come--let's see. What would this be? This would be 9 plus 9; it'd be 18/5. That would be 18/5, which would be what? 3 and 3/5, which is 3.6. It's a population mean for this population of numbers. If we just rearranged the math here a little bit we could view it a slightly different way. How many 3's do we have? We have three 3's, so we could view this as 3 times 3. How many 4's do we have? We have one 4, so it's plus 1 times 4. Plus 1 time 5. All of that divided by 5. And then what we could do is, this is the same thing. And I'm just doing a little bit of basic really, number manipulation here. This is the same thing as 1/5 times 3 times 3 plus 1 times 4 plus 1 times 5. And if we distribute this 1/5 this is equal to what? 3/5 times 3 plus 1/5 times 1. So it's plus 1/5 times 4 plus 1/5 times 5. And this we can see in a bunch of different ways. Let me express these as decimals. So 3/5. So it's 0.6 times 3 plus 0.2 times 4 plus 0.2 times 5. Or we could express these decimals as percentage. We could say 60% times 3, plus 20%-- sorry, 20% times 4. Plus 20% times 5. This is identical to adding up the numbers and then dividing by the total number of numbers there are. But this is interesting because here we had to know how many total numbers there are. We have to say, OK, we added up 5 numbers, we divided by 5. All I did is I played around with the arithmetic a little bit. And I got to this expression. But this expression's more interesting, or at least it's-- well, it's different. It's not necessarily more interesting. I don't want to make any value judgment about it. But here I don't know how many numbers there are. I'm just telling you about the frequency of the numbers. I'm telling you that 60% of the numbers are 3, 20% of the numbers are 4, and then 20% of the numbers are 5. And then if I were to calculate this out I would get 60% times 3 is 1.8. Plus 20% times 4 is 0.8. Plus 20% times 5 is-- let's see. 20% plus 1, which would be equal to 2.6, plus 1 is equal to 3.6. So we would get the exact same number, but what's interesting here is that this tells you just the frequencies-- really the relative frequencies of the 3's, the 4's, and the 5's. What percentage of this population is 3's? 60%. What percentage is 4's? 20%. And what population is 5's? And I'm doing that because we just talked about random variables and all of that. In the beginning we started our statistics discussion about populations and samples. But if you think about it, every time you do one of your experiments and you get a new value for a random variable-- let's do our classic example. We have our random variable, x, is equal to-- I don't know-- it's equal to the number of heads after 6 tosses of a fair coin. So that's our random variable. Hopefully now, we can kind of connect what we thought about in terms of just arithmetic mean and central tendency and population versus sample, and then connect that to the notion of a random variable. So when we first started talking about statistics we said, OK, you have this notion of a population. And that you would sample the population. And we gave a couple of examples. You know, the most common one is you wanted up predict the outcome of a presidential election. The population is everyone who's going to vote in the election. You can't survey all 50 million people or whatever's going to vote for the election. So what you do is you survey a random sample of that population and then you can calculate statistics on that sample that hopefully can estimate the population as a whole. But what happens if the population is not finite? And just to go back, if the population is finite you can calculate things like the population mean. We learned the population mean was that mu letter. And that was you just literally take up all of the items in the population, add them together, and you divide by the number of items there are. That's what we did up here. If this was a whole population of numbers, we've figured out that this was mu. If this was a sample from a population then this would be the sample mean, but we learned all about that. But that's not what I want to get at now. But what happens if this population is infinite? If it's infinite and you're like, oh, Sal, that doesn't make any sense. But if you think about it, well, a random variable really is-- you can kind of view it as each instance of a random variable. Or every time you performed the experiment you're taking out an instance of an infinite population. You can perform this experiment an infinite number of times. You can just keep doing it. It's not like, after doing it a thousand times you're like, oh, you can't toss a coin six times anymore and count the number of heads. You can perform this indefinitely. So every specific result from a random variable-- and those are usually lowercase results, lowercase x1 or x2 or x3. These are just specific instances of a random variable. You can view these as samples from an infinite population. So I'll try to draw an infinite population; it's kind of harder. Maybe I'll draw arrows that go off in every direction. This population never ends. You can keep performing the experiment and keep getting samples, but you're sample is usually finite. You know, let's say we performed this experiment. We toss a fair coin six times and we do that experiment-- I don't know-- we do it a hundred times. So then we would have a hundred samples, x2 and it would go all the way to x100. And the reason why I'm doing this connection is one, to make you see the connection between the random variable and the probability, and the statistics that we talked about earlier. And in this video, I'm going to introduce you to the concept of the expected value of a random variable. And it's nothing else. So the expected value of a random variable, the expected value of a random variable is the exact same thing as the population mean. In fact, sometimes it's called a population mean. But what makes it interesting is in this situation, you have an infinite population. So you can't just add up all the numbers and divide by the number of numbers you have because you have an infinite number of numbers. But what you can do is if you said wow, I know the frequency of the numbers. I know that 3 shows up 60% of the time, 4 shows up 20% of the time, 5 shows up 20% of the time. Then, even if you have an infinite number of numbers, you can actually still calculate a mean. And that's how you do it for an expected value of a random variable. So how do you figure out the frequencies that numbers show up? Well, you can look at the probability distribution, the discreet probability distribution. So in that example that we did last time, I forgot the exact numbers, but actually let me just take what we did-- our Excel out. Let me just quickly, I want to make n 6 trials, probability of heads, tails-- OK, 0.5. And then I need to change what this chart-- just give me one second. Change what the inputs of this chart are. I'm off the screen right now. OK, there you go. So this is the probability distribution for what I just described. I have a fair toss of a coin and I want to know how many heads I have after 6 tosses. So you can perform this experiment a bunch of times, but this tells you the frequency, the frequency of that random variables. So when you perform that experiment-- let's see, whatever this is-- 0.09% of the time. No, actually, 9% of the time you're going to get exactly 1 head. 23% of the time you're going to get 2 heads. 31% of the time you're going to get 3 heads. 23% of the time you're going to get 4 heads. And then, you know 9% of the time you're going to get 5 heads. And then 2% of the time you're going to get 6 heads. So if you have that information you can then actually figure out the population mean for this population that's described by this probability distribution, or the expected value. And let's do that right here. I'll put this over to the side. So I'm looking at that chart I just did while I do this. We just looked at the probability distribution for this random variable, the number of heads after 6 tosses of a fair coin. So the expected value of our random variable is going to be each outcome. So the first outcome is that we had 0 heads times the frequency that 0 shows up. So we figured out before that the frequency-- now, it's a little inexact because I don't have-- actually, I have the exact numbers. 0 will show up in our random variable 0.01563% of the time. So let me write that. So we're going to say, well, this happens and I can write it as a percentage 1.563% of the time. Plus 1 happens 9.375% of the time. And then plus 2 happens 23.438% of the time. Plus 3 happens-- let's see, it says 31.25% of the time. Almost there. 4, I get 4 heads out of 6 tosses 23% percent of the time. So times 23.438%. I get 5 heads after 6 tosses 9.375% of the time. And finally, I get all heads let's see-- no, I get all heads 1.563% of the time. And that makes sense again because all heads should be just as likely as all tails. All tails is the same thing is no heads. So what we did here is exactly what we did up here. We took the relative frequency of each of the numbers in the population and we multiply that outcome times its relative frequency, and we're adding it up. But this is the exact same thing mathematically as we did up here. But what's useful now is we can apply the same principles, but we're finding the arithmetic mean of an infinite population, or the expected value of a random variable, which is the same thing as the arithmetic mean of the population of this random variable. So this value would be equal to-- actually, let me just use Excel to calculate it. So the expected value of getting-- the number of heads you get after 6 tosses. So this is 0 times its frequency, and then I'm going to add them all up and then just will just do that same thing in all of it. So this says this will be 1 times its frequency, 2 times its frequency, and then if I were to take the sum of all of them-- equals sum of all of these-- I get exactly 3. And that's actually kind of an expected outcome, right? I shouldn't use the word expected too much. That the central tendency or you could say, the population mean, of this random variable, or you could say the expected value of this random variable is exactly 3. And in this example it turned out that 3 is also in kind of the colloquial sense, it's the most expected value. It's the most probable value. But we'll see in the future that the expected value doesn't have to be the most probable value. You could have a very high probability of having no heads and a very high probability of having 6 heads. And then you'd still have an expected value of 3, even if 6 or 0 were more probable. And I'll show you more examples of that. But the purpose of this video is to really show you that the expected value calculation is the same thing as the population mean calculation, but we do it this way because you can't add up an infinite number of data points and divide by an infinite number. Instead you want to know the frequencies of each of the outcomes and then you just add up all the outcomes weighted by their frequencies. But that's no different than what you did up there. And I really want to hit that point home because sometimes in probability books they'll just give you a formula-- oh, the expected value of a probability distribution is each of the outcomes times their frequency. But I want to show you that that is the same thing as the population mean. Anyway, see you in the next video.

Contents

Definition

Finite case

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.

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.
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 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 of which are equally likely with a probability of 1/6. 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 1/38 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.

Example

  • 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

If is a random variable whose cumulative distribution function admits a density , then the expected value is defined as the following Lebesgue integral:

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.

General case

In general, if is a random variable defined on a probability space , then the expected value of , denoted by , , or , is defined as the Lebesgue integral

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 ,

Basic properties

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 , .

Linearity

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

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

Monotonicity

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 .

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 ).

Extremal property

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.

Non-degeneracy

If , then (a.s.).

If then (a.s.)

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.

Non-multiplicativity

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

and

The amount by which the multiplicativity fails is called the covariance:

However, if 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

Countable non-additivity

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

Inequalities

Cauchy–Bunyakovsky–Schwarz inequality

The Cauchy–Bunyakovsky–Schwarz inequality states that

Markov's inequality

For a nonnegative random variable and , the Markov's inequality states that

Bienaymé-Chebyshev inequality

Let be an arbitrary random variable with finite expected value and finite variance . The Bienaymé-Chebyshev inequality states that, for any real number ,

Jensen's inequality

Let be a Borel convex function and a random variable such that . Jensen's inequality states that

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.

Lyapunov's inequality

Let . Lyapunov's inequality states that

Proof. Applying Jensen's inequality to and , obtain . Taking the th root of each side completes the proof.

Corollary.

Hölder's inequality

Let and satisfy , , and . The Hölder's inequality states that

Minkowski inequality

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

Fatou's lemma

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).

Corollary. Let

  • pointwise (a.s.);
  • for some constant (independent from );
  • (a.s.), for every

Then

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,

where

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 .

The mass of probability distribution is balanced at the expected value, here a Beta(α,β) distribution with expected value α/(α+β).
The mass of probability distribution is balanced at the expected value, here a Beta(α,β) distribution with expected value α/(α+β).

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].

Expected values can also be used to compute the variance, by means of the computational formula for the variance

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 :

This formula also holds in multidimensional case, when is a function of several random variables, and is their joint density.[5][6]

Alternative formula for expected value

Formula for non-negative random variables

Finite and countably infinite case

For a non-negative integer-valued random variable

General case

If is a non-negative random variable, then

and

where denotes improper Riemann integral.

Formula for non-positive random variables

If is a non-positive random variable, then

and

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

General case

If can be both positive and negative, then , and the above results may be applied to and separately.

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. 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.[7]

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/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:

… 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,[8] 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".[9]

See also

References

  1. ^ 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.
  2. ^ 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.
  3. ^ 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!
  4. ^ 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.
  5. ^ Expectation Value, retrieved August 8, 2017
  6. ^ Papoulis, A. (1984), Probability, Random Variables, and Stochastic Processes, New York: McGraw–Hill, pp. 139–152
  7. ^ "Ore, Pascal and the Invention of Probability Theory". The American Mathematical Monthly. 67 (5): 409–419. 1960. doi:10.2307/2309286.
  8. ^ Whitworth, W.A. (1901) Choice and Chance with One Thousand Exercises. Fifth edition. Deighton Bell, Cambridge. [Reprinted by Hafner Publishing Co., New York, 1959.]
  9. ^ "Earliest uses of symbols in probability and statistics".

Literature

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