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I have been going through the definition of expected value in Wikipedia (http://en.wikipedia.org/wiki/Expected_value) beneath all that jargon it seems that the expected value of a distribution is the average value of the distribution. Did I get it right ?

If yes, then what is the point of introducing a new term ? Why not just stick with the average value of the distribution ?

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  • $\begingroup$ Average is essentially expected value. Except I tend to use (IMHO ) average for simple average I.e expected value of X say, whilst expected value is usually a function. But your right, for example average waiting time, would in my head, be also the expected time to wait. As long as you know what you mean. Though some stats guys and mathematicians will properly disagree. $\endgroup$ – Chinny84 Aug 20 '14 at 18:43
  • $\begingroup$ @Chinny84 Thanks for the clarification. Also, does "Central tendency" convey the same sense as average ? $\endgroup$ – user2340452 Aug 20 '14 at 18:49
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    $\begingroup$ Expected value is very much like an average, but we mostly use the term "average" for finite sets of values, while "expected value" can be used more broadly than that. It is still useful to think of "expected value" as an average. $\endgroup$ – Thomas Andrews Aug 20 '14 at 19:01
  • $\begingroup$ No, I am not sure. Will delete. $\endgroup$ – André Nicolas Aug 20 '14 at 21:47
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    $\begingroup$ Related question here on Cross Validated. $\endgroup$ – StubbornAtom Jan 3 '17 at 6:42
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From my experience so far in statistics, I have more often heard "average" when discussing samples and in nonparametric statistics. I have first seen the definition of the expected value in a frequentist parametric statistic context, and we understood the expected value as the average of the outcomes when repeatedly repeating the procedure (the average is an unbiased estimator of the mean), which is basically the average you are discussing.

Hence, often, when the average is discussed, we mean the sample average (funny word play there). We compute the sample average on a given set of random variables (sample), that is a set of outcomes of a distribution. This average may yield different properties with regards to the estimation of the "actual average" of the underlying distribution, for instance you may consider how the mathematical definition of the sample average behaves when passing to the limit (taking the sample size to infinity), etc.; but the expected value is functionally associated to distribution with a given parameter,- a distribution that can further generate samples with different sample averages.

Suppose $X_1,X_2,...,X_n$ is a sample of i.i.d. random variables. Observe that we have, in general $$\frac{\sum_{k=1}^nX_k}{n}\neq E(X_i).$$

The terms are used interchangeably, but one must be careful with what exactly is being discussed.

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  • $\begingroup$ So .. Average -> sample and (expected value) -> Distribution ?? That is the convention usually ? $\endgroup$ – user2340452 Aug 20 '14 at 18:52
  • $\begingroup$ Sorry, I will edit the above to be more clear. $\endgroup$ – EkS Aug 20 '14 at 18:55
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    $\begingroup$ I'd say "average" can refer to a sample, so "expected value" is often clearer when indicating that we are talking about the entire random variable. $\endgroup$ – Thomas Andrews Aug 20 '14 at 19:04
  • $\begingroup$ I have completed my answer above. I hope I am clear now, please respond with your own thoughts and experience, I am interested to know if people generally agree with the above. $\endgroup$ – EkS Aug 20 '14 at 19:08
  • $\begingroup$ Your reply kinda makes sense to me now. Along with @ThomasAndrews comment. $\endgroup$ – user2340452 Aug 20 '14 at 19:15
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The concept of expectation value or expected value may be understood from the following example. Let $X$ represent the outcome of a roll of an unbiased six-sided die. The possible values for $X$ are 1, 2, 3, 4, 5, and 6, each having the probability of occurrence of 1/6. The expectation value (or expected value) of $X$ is then given by

$(X)\text{expected} = 1(1/6)+2\cdot(1/6)+3\cdot(1/6)+4\cdot(1/6)+5\cdot(1/6)+6\cdot(1/6) = 21/6 = 3.5$

Suppose that in a sequence of ten rolls of the die, if the outcomes are 5, 2, 6, 2, 2, 1, 2, 3, 6, 1, then the average (arithmetic mean) of the results is given by

$(X)\text{average} = (5+2+6+2+2+1+2+3+6+1)/10 = 3.0$

We say that the average value is 3.0, with the distance of 0.5 from the expectation value of 3.5. If we roll the die $N$ times, where $N$ is very large, then the average will converge to the expected value, i.e.,$(X)\text{average}=(X)\text{expected}$. This is evidently because, when $N$ is very large each possible value of $X$ (i.e. 1 to 6) will occur with equal probability of 1/6, turning the average to the expectation value.

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The expected value, or mean $\mu_X =E_X[X]$, is a parameter associated with the distribution of a random variable $X$.

The average $\overline X_n$ is a computation performed on a sample of size $n$ from that distribution. It can also be regarded as an unbiased estimator of the mean, meaning that if each $X_i\sim X$, then $E_X[\overline X_n] = \mu_X$.

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The distinction is subtle but important:

  1. The average value is a statistical generalization of multiple occurrences of an event (such as the mean time you waited at the checkout the last 10 times you went shopping, or indeed the mean time you will wait at the checkout the next 10 times you go shopping).
  2. The expected value refers to a single event that will happen in the future (such as the amount of time you expect to wait at the checkout the next time you go shopping - there is a 50% chance it will be longer or shorter than this). The expected value is numerically the same as the average value, but it is a prediction for a specific future occurrence rather than a generalization across multiple occurrences.
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Mean or "Average" and "Expected Value" only differ by their applications, however they both are same conceptually.

  • Expected Value is used in case of Random Variables (or in other words Probability Distributions). Since, the average is defined as the sum of all the elements divided by the sum of their frequencies. But for the case of Probability distribution we can't describe a random variable in terms of its frequency beforehand, thus we use the probability instead. Conceptually, probability of an element is frequency of an event divided by size of sample space. Thus, the average in case of random variable can be given by sum of probabilities multiplied by its respective event (where p(x)*x is conceptually frequency of x divided by total frequency).

  • Average on the other hand is used in case where we have the knowledge of frequencies of individual elements and total count of the elements, for example, in case of known data set or sample. We can simply use the fundamental definition of average to calculate it.

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