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

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 $$\zeta$$ ?

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 $$x_1,x_2,\ldots,x_k$$".
• @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. – grand_chat Oct 11 '18 at 7:13