What is the difference between Average and Expected value?

Question

Question : What is the difference between Average and Expected value?

---

I have been going through the definition of expected value on Wikipedia 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 ?

Answer 1

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.

Answer 2

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 $E_X[\overline X_n] = \mu_X$.

Answer 3

I think to better understand a concept is good to know the motivation behind the creation of the concept. This make the concept 'alive'. There was this guy called Chevalier de Méré, a French nobleman, who brings to Pascal a problem about a game:

A team plays ball such that a total of $60$ points is required to win the game, and each inning counts $10$ points. The stakes are $24$ ducats. By some incident, they cannot finish the game when one side has $50$ (team A) points and the other $30$ (team B).

So, how should the prize be divided?

Pascal starts to write to Fermat about this problem. At first, people argued that the fair way to divide would be for example the aristotle's proportional division: $50:30$ or $5 \text{ to } 3$.

But that didn't seem fair, because team A needed only to win one more game in comparison with team B. None of the solutions seemed fair.

So Fermat and Pascal concluded something groundbreaking: to think about the future.

What is expected to happen?

Well, Team A could win in the next game or could lose, and the next one:

So the probability of team A winning is $\frac 78$, and the expected value of team A to receive is $\frac 78 \cdot 24 \text{ ducats } = 21 \text{ ducats }$. So the fair division if the game was not interrupted would be $3$ ducats to team B and to team A $21$ ducats. So this opened up the possibility of mathematicians predicting the future. This created industries like insurance and many others.

Well about the average there are many averages:

In colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value. Source

So if you are talking about the arithmetic mean, I would say the arithmetic mean is not weighted by the respective probability of an event happen. You just sum all the elements and divide the sum by the total quantity. This should be the most representative number and which equilibrates your set of numbers. Also, with arithmetic mean we are not talking about the future like expected value.

Answer 4

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.

Answer 5

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.