To probability lovers, I just want to share (and discuss) few simple problems I solved in Chapter 4 of Casella, G. and Berger, R.L. (2002). Statistical Inference.

A random point $(X,Y)$ is distributed uniformly on the square with vertices $(1, 1),(1,-1),(-1,1),$ and $(-1,-1)$. That is, the joint pdf is $f(x,y)=\frac{1}{4}$ on the square. Determine the probabilities of the following events. $X^2 + Y^2 < 1$$2X-Y>0$$|X+Y|<1$ (modified since the original $|X+Y|<2$ is trivial.)Solutions:$X^2 + Y^2 < 1$

We need to consider the boundary of this inequality first in the unit square, so below is the plot of $X^2 + Y^2 = 1$,

A random point $(X,Y)$ is distributed uniformly on the square with vertices $(1, 1),(1,-1),(-1,1),$ and $(-1,-1)$. That is, the joint pdf is $f(x,y)=\frac{1}{4}$ on the square. Determine the probabilities of the following events. $X^2 + Y^2 < 1$$2X-Y>0$$|X+Y|<1$ (modified since the original $|X+Y|<2$ is trivial.)Solutions:$X^2 + Y^2 < 1$

We need to consider the boundary of this inequality first in the unit square, so below is the plot of $X^2 + Y^2 = 1$,