### On the expected difference between mean and

Forums New posts Unanswered threads Search forums. Best Answers. Media New media New comments Search media. Groups Search groups Upcoming events. Log in Register. Search titles only.

Arithmetic mean is simple average of numbers not weighted by anything. However in Expected Value, the numbers are weighted by their probability. Here the mean is 3. Here again answer is 3. Actually when we multiply by weight we get our data back and we essentially do the same thing.

So I again ask is there any difference between the two. Assume you have the following: two, 1's; ten, 2's; and one, By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. On Halloween trick-or-treaters arrive at a house from pm until 9pm according to a Poisson Process with an average of 12 trick-or-treaters per hour.

But how do we do the second part?

4. Expected Value and the Meaning of the Mean

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. What is the difference between expected value and 'most likely' value? Ask Question. Asked 5 years, 4 months ago. Active 5 years, 4 months ago. Viewed 5k times. L L 3 3 silver badges 12 12 bronze badges. The most likely value is the value that will appear the largest number of times. The most likely value is 2. The most likely value is the mode; the most probable value.

## â€śAverage,â€ť â€śMean,â€ť â€śMedian,â€ť And â€śModeâ€ť: Whatâ€™s The Difference?

Feedback post: New moderator reinstatement and appeal process revisions. The new moderator agreement is now live for moderators to accept across theâ€¦.

The unofficial elections nomination thread. Hot Network Questions. Question feed. Mathematics Stack Exchange works best with JavaScript enabled.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. What is the precise difference between the Mean, Average and Expectation? Is there any mathematical formaulation to show the difference? I see that often they are used interchangeably.

They are pretty much interchangeable. Which you use and when may have different connotations, however. And while almost everyone recognizes them as interchangeable, some people and textbooks do not agree. Here are their connotations, as far as I know:. There already is an answer on CV regarding the difference between mean and averagewhich actually ends at saying that, briefly, a mean is a particular case of average. Whitout entering too much in details since Wikipedia does it very wellhere are some intuitions:.

An expected value is a theoretical average in the sense that you need to get all observations related to the phenomenon under question to compute it, which is rarely achievable, too costly, or simply impossible. This leads to approximate the expected value instead of really getting it.

Note that the more data you have, the better its approximation, since this approximation is said, hopefully, to converge of the sample size: this is the so-called Asymptotic law.

Which commonly works very well, e. TimAtreides' flipping coin perfectly illustrates this point. And sometime, even if you have a big sample size, and that you can compute an average, the obtained value has no real sense, because the black box process which has generated your data is governed by a law of distribution which has simply no Expected value. Imagine, for example, that you have the following dataset. For sure, you will be able to compute the average. Your average is very likely to change radically.

The data generating process we are trying to explain follows a distribution which has no expected value. A canonical example of this is the Cauchy distribution.

Sometimes what one computes is a not so bad approximation of what is going out from the black box, sometimes not. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Mean, Average and Expectation Ask Question. Asked 3 years, 2 months ago.

Active 3 years, 2 months ago. Viewed 2k times. Active Oldest Votes. Here are their connotations, as far as I know: Mean : The arithmetic mean. That is, the sum divided by the total number of observations. Not to be confused with the geometric mean. Average : Almost always the same as the mean.

Some people believe the term "average" to be the mean paired with its standard deviation. Some people consider any measure of center to be an average. But the people who use this context are small. Don't know if I've ever met any in real life.This means that over the long term of doing an experiment over and over, you would expect this average. You toss a coin and record the result. What is the probability that the result is heads?

### Difference Between Average and Mean

If you flip a coin two times, does probability tell you that these flips will result in one heads and one tail? You might toss a fair coin ten times and record nine heads. Probability does not describe the short-term results of an experiment. It gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24, times!

He recorded the results of each toss, obtaining heads 12, times. In his experiment, Pearson illustrated the Law of Large Numbers. The Law of Large Numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero the theoretical probability and the relative frequency get closer and closer together.

In other words, after conducting many trials of an experiment, you would expect this average value. The probability that they play zero days is 0.

X takes on the values 0, 1, 2. In this column, you will multiply each x value by its probability. Expected Value Table. This table is called an expected value table. The table helps you calculate the expected value or long-term average.

The expected value is 1. The number 1. Calculate the standard deviation of the variable as well. The fourth column of this table will provide the values you need to calculate the standard deviation. For each value xmultiply the square of its deviation by its probability.

A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a hour shift. For a random sample of 50 patients, the following information was obtained.

What is the expected value? Suppose you play a game of chance in which five numbers are chosen from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

A computer randomly selects five numbers from zero to nine with replacement. Over the long term, what is your expected profit of playing the game? The values of x are not 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

To win, you must get all five numbers correct, in order. You may choose a number more than once. The probability of choosing all five numbers correctly and in order is.Indeed, every teacher is expected to be a Muslim by birth or conversion. He is expected to spend the next few days closeted with lawyers and advisers at his home, Royal Lodge, in Windsor Great Park. He expected European capitalism to evolve spontaneously into a market socialism of worker-owned cooperatives.

Essentially, we are being left in a position where we are expected to just take agency promises at face value. That he had some notice of what was to be expected from that quarter, appears by the following letter to his friend, Mr. Somehow he had expected to find her there, and he watched her again, as he had done through Pre Antoine's vines. What else could have been expected than discomfiture and disgrace?

I turned my eyes in the direction I expected to see the horses. I expected a blow for that, and tried to look as though I did not, being extremely anxious to return it with effect. I expect him later. She expects that they will come. If so, see if you can dig up the meanings of these vocab words from the popular novel. Idioms for expect be expectingto be pregnant: The cat is expecting again.

Expect implies confidently believing, usually for good reasons, that an event will occur: to expect a visit from a friend. Anticipate is to look forward to an event and even to picture it: Do you anticipate trouble? Hope implies a wish that an event may take place and an expectation that it will: to hope for the best.

Await wait for implies being alert and ready, whether for good or evil: to await news after a cyclone. This sense of expect I expect you went with them.

I expect you want to leave now. Words nearby expect expansivityexpatexpatiateexpatriateexpatriationexpectexpectancyexpectantexpectationexpectation of lifeexpectation sunday. Words related to expected normalscheduledlikelyproposedcomingforeseencontemplatedenvisionedawaitedforetoldprophesiedconventionalfamiliarhabitualduepredictableforeseeable. Example sentences from the Web for expected Indeed, every teacher is expected to be a Muslim by birth or conversion.

Life of Lord Byron, Vol. Thomas Moore. Bayou Folk Kate Chopin. Snow Shoes and Canoes William H. Some Persons Unknown E. See also expecting. Derived forms of expect expectableadjective expectablyadverb. Test Your Vocab Skills Now!In probability theorythe expected value of a random variable is a generalization of 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 expectationmathematical expectationmeanaverageor first moment. Expected value of a general random variable involves integration in the sense of Lebesgue. The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of pointswhich seeks to divide the stakes in a fair way between two players who have to end their game before it's properly finished.

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. 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. In this sense this book can be seen as the first successful attempt at laying down the foundations of the theory of probability. 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.

## Subscribe to RSS

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. 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.

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.Related to expected: expected value. To look forward to the probable occurrence or appearance of: expecting a phone call; expects rain on Sunday. To consider likely or certain: expect to see them soon. See Usage Note at anticipate. To consider reasonable or due: We expect an apology. To consider obligatory; require: The school expects its pupils to be on time.

To be pregnant. Used in progressive tenses: My wife is expecting again. Synonyms: expectanticipatehopeawait These verbs relate to the idea of looking ahead to something in the future. To expect is to look forward to the likely occurrence or appearance of someone or something: "We should not expect something for nothingâ€”but we all do and call it Hope" Edgar W. Anticipate sometimes refers to taking advance action, as to forestall or prevent the occurrence of something expected or to meet a wish or request before it is articulated: anticipated the storm and locked the shutters.

The term can also refer to having a foretaste of something expected: The police are anticipating trouble with rowdy fans after the game. To hope is to look forward with desire and usually with a measure of confidence in the likelihood of gaining what is desired: I hope to see you soon.

To await is to wait expectantly and with certainty: She is eagerly awaiting your letter. Switch to new thesaurus. Known to be about to arrive: anticipatedduescheduled. Mentioned in?

References in classic literature? But however prolonged and exhausting the chase, the harpooneer is expected to pull his oar meanwhile to the uttermost; indeed, he is expected to set an example of superhuman activity to the rest, not only by incredible rowing, but by repeated loud and intrepid exclamations; and what it is to keep shouting at the top of one's compass, while all the other muscles are strained and half started --what that is none know but those who have tried it.

View in context. As they entered the yard Leslie came out on the veranda from the side door, peering through the gloom for some sign of her expected guest.

THE author of the "Notes on the State of Virginia," quoted in the last paper, has subjoined to that valuable work the draught of a constitution, which had been prepared in order to be laid before a convention, expected to be called inby the legislature, for the establishment of a constitution for that commonwealth. The father and mother did not speak of the matter to their son again, but a few days later the countess sent for Sonya and, with a cruelty neither of them expectedreproached her niece for trying to catch Nicholas and for ingratitude.

The bridegroom was expected at the church while he was pacing up and down his room like a wild beast in a cage, peeping out into the corridor, and with horror and despair recalling what absurd things he had said to Kitty and what she might be thinking now.

Every day at Longbourn was now a day of anxiety; but the most anxious part of each was when the post was expected. The arrival of letters was the grand object of every morning's impatience. And all the time, deliriously happy, talking in the gentlest and most delicate terms that might not offend her sensitive ears, I was cudgelling my brains in an effort to divine what I was expected to do.

It is disheartening, for one naturally expects to find anything we seek a few minutes after starting out to seek it, and I confess that I expected to find my golden mistress within a very few hours of leaving home. As this is one of those deep observations which very few readers can be supposed capable of making themselves, I have thought proper to lend them my assistance; but this is a favour rarely to be expected in the course of my work.

Dashwood could spare them perfectly well; and Elinor, who now understood her sister, and saw to what indifference to almost every thing else she was carried by her eagerness to be with Willoughby again, made no farther direct opposition to the plan, and merely referred it to her mother's decision, from whom however she scarcely expected to receive any support in her endeavour to prevent a visit, which she could not approve of for Marianne, and which on her own account she had particular reasons to avoid.

She had seen a good deal of the Gardners; she and Dorothy were very intimate; college circles expected the announcement of her engagement to Roy any day.