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See this link at Page-$07$.

If the stochastic process is discrete-valued, then a collection of probability mass functions can be used to specify the stochastic process:

$$P_{X_1···X_k}(x_1, x_2, · · · , x_k) = P(X_1 = x_1, X_2 = x_2, ··· , X_k = x_k)$$

How does the above r.p. notation read in plain English?

As far as I can tell:

Probability of the Random Process $X(k, \zeta)$ is equal to the joint probability of the random variables $X_1, X_2, ... , X_k$.

Note. $\zeta$ is the sample space.

But, why is it written as $P(X_1 = x_1, X_2 = x_2, ··· , X_k = x_k)$?

It seems that $X_1$ can't take the value $x_2$? Why?

Aren't $x_1$ and $x_2$ both members of sample space $\zeta$ ?

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The lower-case $x_1, x_2, \ldots, x_k$ are dummy variables. They represent generic values for the corresponding random variables $X_1, X_2,\ldots, X_k$, the same way that $x$ and $y$ are generic variables in the joint density $f(x,y)$ of a bivariate pair $(X,Y)$. The dummy variables have different subscripts so we can keep straight which dummy variable is associated with which random variable.

In equations involving $x_1,x_2,\ldots,x_k$, unless explicitly restricted, these variables are understood to take any legal value: For example, in the equation (which is actually defining the LHS) $$ P_{X_1···X_k}(x_1, x_2, · · · , x_k) = P(X_1 = x_1, X_2 = x_2, ··· , X_k = x_k) $$ you should read the implicit "for all possible values of $x_1,x_2,\ldots,x_k$".

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  • $\begingroup$ Are rest of my conceptions correct? $\endgroup$ – user366312 Oct 11 '18 at 7:08
  • $\begingroup$ @yahoo.com Yes, the probability mass function $P_{X_1···X_k}(x_1, x_2, · · · , x_k)$ is shorthand for the joint probability $P(X_1 = x_1, X_2 = x_2, ··· , X_k = x_k)$. The assertion is that to define the random process, it is enough to specify the collection of p.m.f.'s as $k$ ranges over all positive integers. $\endgroup$ – grand_chat Oct 11 '18 at 7:13

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