Indicator Function Notations and Joint Probability Density Function I have the following exercise: 
Let the Joint PDF be given as $f(x,y) = I(0 < x <2)I( x <2(1-y))$. Find $P(X > 1, Y < \frac{1}{4})$ and $P(X > Y)$.
Firstly, I do not understand this indicator function notation. To my understanding, the indicator function tells me the domain of $x$ or $y$, but in this case they are using it to define the function. So then what is the function? 
Picture of Question

In general indicator function $I(a\leq x \leq b)$ is just defined $I(x)=1$ for $x \in [a,b]$ and $I(x) = 0$ otherwise. 
So your function $f(x,y)=1$ for pairs $(x,y) \in A$, 
$$A= \{(x,y) \in \mathbb R^2: x \in (0,2), x < 2(1-y) \}.$$
Otherwise  $f(x,y)=0.$
Can you now count what is needed?
Edit:
Now we seen that 
$$A = \{(x,y) \in \mathbb R^2: x \in (0,2), y \in \left(-\infty,1-\frac{x}{2} \right) \}.$$
So if $f$ is density function then must be true that 
$$\int_0^2 \int_{-\infty}^{1-x/2} 1 dydx=1,$$
which is evidently not correct, the integral is not finite. Did you rewrite the exercise correctly?
Also just a note, when you evaluate integral in $\mathbb R^2$ you always start to integrate from the most inner integral. Therefore the order is important, see wiki
