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I don't understand what stationarity of random process mean.
I know they're statistical properties that are time invariant but I don't have intuition for it and I don't get what that has to do with random process joint probability density function intuitively.
Why is a random process strictly stationary when its joint probability density function is time invariant?
This query, taken from the title of the question cannot be answered because there is no Why.
If for each integer $n\geq 1$ and time instants $t_1, t_2, \ldots, t_n$ and $\tau$, it so happens that the (joint) distribution of $X(t_1), X(t_2),\ldots, X(t_n)$ is the same as the (joint) distribution of $X(t_1+\tau), X(t_2+\tau), \ldots, X(t_n+\tau)$ -- that is, moving the $n$ time instants $t_1, t_2, \ldots t_n$ to the right by $\tau$ does not change the distribution of the variables -- then the process is called a (strictly) stationary process. There is no intuition involved in the matter. Someone came up a long time ago with the name stationary as shorthand for the case when a random process has the property that its joint density functions are time-invariant, and other people found the concept useful and the name useful. They found that the chosen name was apt (stationary, meaning unmoving, reminds us that the distribution is not changing), easy to remember, and less of a mouthful than the long-winded "joint distributions are invariant to time shifts" etc. And so the name stuck.
If you would like to understand more about random processes, stationarity etc, consider reading this answer of mine on this forum.
