This doubt is from the book Stochastic Processes by Ross..Chapter 1, subsection Random variables.
The distribution function F of the random variable X is defined for any real number x by
F(x) = P{X$\leq$x} = P{X $\epsilon$ (-$\infty$,x]}
I have not understood why X $\epsilon$ (-$\infty$,x] and not [0,x]
How is X $\epsilon$ (-$\infty$,x] to be practically interpreted? Request help understand
Let us consider a particular example. Suppose that a random variable $X$ is equal to $-1$ almost surely. What is the probability that the value of this random variable is less than or equal to some positive number $x$, i.e. $P(X\le x)$, where $x>0$? This probability is not equal to the probability $P(X\in[0,x])$, it is equal to the probability $P(X\in(-\infty,x])$.
I hope this helps.