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13

Just suppose I give you a series of numbers, and I tell you they were picked randomly. And you know I am not trying to deceive you. Numbers are: $3$, $1$, $4$, $1$, $5$, $3$, $2$, $3$, $4$, $3$. I now propose you to predict the next one, or at least, to be as close as possible. Which number would you pick? [Think] [Compute] I bet most of the readers ...


9

From the wikipedia article: a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process. In other words: the time-ensemble statistical properties are the same as the realization-ensemble statistical properties. Maybe we need to take a step back and talk about what ...


7

Let us consider a hypothetical random process where the sample functions are DC values and are different from each other: X1(t) = constant= mean of X1(t) X2(t) = constant= mean of X2(t) The temporal mean of $X_1(t)$ and $X_2(t)$ are constant but not equal. if my process is stationary $X(t_1)$ and $X(t_2)$ are equal and RVs (refer Dilip's answer) So ...


5

You can compute the power of the process from its power spectrum as well as from its PDF. Equating the two gives you a relation between the constants $A$ and $B$. More specifically you get $$\int_{-B}^BG_x(f)df=\frac{1}{2A}\int_{-A}^Ax^2dx$$ If I'm not mistaken this should give $A=\sqrt{3B}$.


4

If you sample a random process for a specific t, you will get one realization of a random variable. For another t, you get another realization of that random variable. This random variable has its statistics which is almost impossible to learn in real world because not all sample paths are observable. See the brown rectangle in the figure below. That ...


3

It is usually more difficult to understand the non-ergodic case (that is why people look for examples of such processes more often). As an example of ergodic process, let the process $X(t)$ represent repeated coin flips. At each time $t$, we have a random variable $X$ that can choose between $0$ or $1$. If it is a fair coin, then the ensemble mean is $\frac{...


3

Joint behavior cannot always be deduced from individual behavior. For example, if $X$ and $Y$ are (nondegenerate) random variables with finite means, then $P\{X<E[X]\}$ and $P\{Y<E[Y]\}$ both are nonzero. This is because there must be probability mass to the left of the mean (and to the right of the mean too) except in the degenerate case when $X$ has ...


3

I hope this video (from the Florida Institute of Technology. Titled "what is wide sense staionary, strict sense, ergodic signals" by Dr. Ivica Kostanic in his Communications Theory class) from 16:55 could clear your doubts


3

I will try to explain this practically using MATLAB notation. Yet before that I must say the ergodic property sometime is limited to a level of moment, namely ergodic in the 1st , 2nd, 3rd moment, etc... If the process is IID it is promised to be ergodic. Now, assume we have a function myRandomProcess in MATLAB. It returns a row vector of length n with the ...


3

Re 2.: ergodicity is always just an assumption, a modeling approach. You logically can't test for it, since that would require knowledge about all realizations over all if eternity. Re 1.: stationarity is a property of a stochastic process, not of one realization. You simply might have had "bad luck" with this realization. But, if you asked me, assuming ...


3

for 1, a random process is WSS if its autocovariance and mean ensemble average don't vary as a function of specific time instance, just lag. As mentioned before, it's hard to conclude this from simply one realization of the process about the ensemble average. However for 2, you can at least estimate what the autocovariance and mean are if the process is ...


2

An eternal well-balanced dice has 1/6 probability for each facet $f$, each time. This uniform probability law yields a mean expectation of 3.5: $\sum_{f=1}^{6} \frac{1}{6}\times f$. Each time you cast the dice, you get this expectation. Of course, for each throw you'll only get an integer 1, 2, 3, 4, 5, or 6, never a decimal like 3.5. So there is an ...


1

Eq. $(2)$ is indeed the periodogram of the truncated signal where said signal is the sample path $x(t)$ of the random process. There is nothing random about this sample path, which is why dropping the Expectation operator makes sense when one goes from $(1)$ to $(2)$. Dropping the limit is asking everyone to take your word for it that the $T$-second ...


1

This is my understanding: the statistics of the source described in the paper depend on which character is produced first. If the first character is $a$, then one of the source properties is that letters in odd positions are always $a$. However, if the first letter is $b$ (in other words, a shift of $1$ in the circuit), then letters in odd positions can be ...


1

As suggested I am adding my comments as an answer. The sample autocorrelation function (ACF) for $n$ observations is given by $\hat{p}_x(m) = \frac{\sum_{n=0}^{N-m-1}(x_n - \bar{x})(x_{n+m} - \bar{x})}{\sum_{n=0}^{N-m-1}(x_n - \bar{x})^2} $ To understand why you have a difference between figure 1 and figure 2, lets assume we are looking at observations $...


1

What the CDF of the capacity tells you is how it distributes, which allows you to say something about the quantiles. Ergodic capacity tells you: what capacity will we see on average? Quantiles are more relevant in practice: what capacity can we guarantee with 95% certainty? 99%? 99.9%? For this, you need a CDF and more often than not, you'll see a heavy-...


1

A WSS Gaussian process is also a strictly stationary Gaussian process, but beyond that, Gaussianity or the lack thereof has nothing to do with the matter; the question might as well have asked for a proof for a general discrete-time stationary process that the process is second-moment ergodic if and only if the following condition holds: $$\lim_{n\to\infty} ...


1

Hi: Just to emphasize what Philip M said. If one assumes ergodicity, then one realization is quite enough to do statistical analysis ( so the checked answer should be modified. ) The problem is that testing for ergodicity is complex and requires more than one realization. Therefore, in practice, most people assume ergodicity and then this allows one to test ...


1

For an example of the opposite case (i.e., a random process that is ergodic but not stationary), consider a white noise process that is amplitude modulated by a deterministic square wave. The time average of of every sample function is equal to zero, as is the ensemble average over all time. So the process is ergodic. However, the variance of any ...


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