Understanding of Random Process, Random Variable and Probability Density Function
when we observe the Random Process at a specific time $t_k$, that is the
  value at $X(s_1,t_k), X(s_2,t_k),\ldots,X(s_n,t_k)$, if we denote them
  by $(a_1,a_2,\ldots, a_n)$. Now the mapping between the outcomes
  $(s_1,s_2,s_3,\ldots,s_n)$ and its probabilities
  $(a_1,a_2,\ldots,a_n)$ are collectively called as Random Variable.


Not true. Random variable is a variable which is ... random where its random characteristic is characterized by its probability density function (PDF). And it is the PDF that is "mapping between the outcomes and its probabilities".

A random process is simply a collection of random variables. $\left\lbrace X(t) \right\rbrace$ is a random process and $X(t_1), ..., X(t_n), ...$ are random variables.

The random variables $\left\lbrace X(t_n) \right\rbrace$ can be identical distributed, i.e. $f_{X(t_1)}(s) = f_{X(t_2)}(s) = \cdots$; or not, i.e. $f_{X(t_1)}(s) \neq f_{X(t_2)}(s) \neq \cdots$ where $f_{X(t_1)}(s)$ denotes the PDF of random variable $X(t_1)$.

Using your notations, $f_{X(t_1)}(s)$ is the probability such that random variable $X(t_1)$ has value $s_1$ of sample space, i.e. $\mathrm{Prob}(X(t_1) = s_1) = f_{X(t_1)}(s_1)$

Update: for the question in comment
Use your notations and if I understand correctly what you mean, for a given time $t=t_k$, we do experiments, and we have outcomes $s_1, \cdots, s_n, \cdots$ which associated probabilities denoted $X(s_1, t_k), \cdots, X(s_n, t_k), \cdots$.

Thus the Probability mass function (or PDF if the support is continous) is $f_{X(t_k)}(s) = \left\lbrace X(s_i, t_k) \textrm{, } s_i \in \left\lbrace s_1, \cdots \right\rbrace \right\rbrace$