I have difficulties in the very beginning of probability theory.
A (discrete) random variable $X$ is a function from a finite or countably infinite sample space $S$ to the real numbers. The probability density function of the random variable $X$: $f(x)=Pr\{X=x\}=\sum_{s\in S:X(s)=x}Pr\{s\}$ If I want to know what probability of $X=x_1\ or\ x_2$, then $Pr\{X=x_1 or X=x_2\}=\sum_{s\in S:X(s)=x_1\ or\ X(s)=x_2}Pr\{s\}=$ Consider that, x1 and x2 are mutually exclusive, I can split the sum $=\sum_{s\in S:X(s)=x_1}Pr\{s\}+\sum_{s\in S:X(s)=x_2}Pr\{s\}=Pr\{X=x_1\}+Pr\{X=x_2\}$ The problem is in the case when there are two independent random variables. How can I show that $Pr\{X=x\ and\ Y=y\}=Pr\{X=x\}Pr\{Y=y\}$ Thanks.