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Yes. For the first problem, determine each of the following sets:
$\{\omega: X(\omega) \le 1\}, \left\{\omega: X(\omega) \le \dfrac{1}{4} \right\}, \{\omega: X(\omega) \le 0\}$
You want to be able to represent each of these sets as an interval or a union of disjoint intervals.
For (b), you are going to want to do the same thing. Represent the set as an interval or union of disjoint intervals.
For (c), $F_X$ is not a notation with which I am familiar. How does your book define that?
For (d), do you really not know how to compute density? That may be something to discuss with your professor.
For (e), same response.
For the second one, the problem is fairly straightforward. I am not sure what sort of suggestion to offer until I know why you are having trouble with it.
That is not always standard notation. However, it is used by Larson&Marx, because Marx is so influential in mathematical statistics it is used in many textbooks. $F_X(t)=\mathcal{P}(X(t)\le t)$ It is commonly called the cumulative distribution function(Cdf).
1) $F_X$ is nondecreasing
2) $F_X$ is continuous on the right.
3) $\displaystyle{\lim _{t \to - \infty }}{F_X}(t) = 0\quad \& \quad {\lim _{t \to \infty }}{F_X}(t) = 1$
4) $\mathcal{P}(X<t)=\bf{\lim _{X \to t^-}}{F_X}(t) $