IT maybe an elementary question but I'm totally new to the concept. In Wikipedia, ergodicity is defined as follows:
In statistics, the term describes a random process for which the time average of one sequence of events is the same as the ensemble average.
Suppose that we have thrown a die one time in 10 consequent days and gotten these results:
$\{5,5,1,5,4,1,2,3,5,5\}$
If we consider $(X_i)_{1\le i\le10}$ the random process for this experiment, in the $i$th day we have 6 possible values for random variable $X_i$ each with probablilty $\frac{1}{6}$ so: $$ensemble\;average=\frac{1+2+3+4+5+6}{6}=\frac{21}{6}=3.5$$
but for temporal average base on this single realization we have:$$temporal\;average=\frac{36}{10}=3.6$$
so tossing a die for 10 subsequent days is not an ergodic random process. But if $number\;of\;days\to\infty$ then $temporal\;average\to 3.5$ so tossing a die for an infinite subsequent days is an ergodic process.
1-Is my conclusion true?
2-Is my understanding for ensemble average, temporal average and ergodicity right?
If yes give me an example of an ergodic and a nonergodic continuous random processes and Compute their ensemble and temporal averages?
If no tell me how should I compute ensemble and temporal averages? should I consider all $6^{10}$ possible realizations when computing temporal average?
Ergodicity is a concept based on the limit of the temporal average. So, based on a small number of experiments you cannot infer that the process is not ergodic.
In the case of independent experiments it can be easily shown what the relationship of the temporal average and the ensemble average. In your case
$$\text{temporal average}=\frac{36}{10}=\frac{5}{10}5+\frac{2}{10}1+\frac1{10}4+\frac1{10}3+\frac1{10}2=3.6$$ Here, say, $\frac5{10}$ is the relative frequency of the outcome $5$. If the number of experiments is large (the temporal length of the experiment is high). Then the estimates of the relative frequencies are good.
It is clear that in the case of independent experiments the temporal average has to tend to the ensemble average. Also, if the estimates of the ensemble probabilities are poor then the temporal average is also poor. Which does not mean the the process is not ergodic.