Two dice are rolled at random. Find the probability distribution of the sum of the numbers on them, find the mean of the random variables.

When two dice are rolled the sample spale “$S$” consists of $6\times 6=36$

$S=\left\{\right(1,1),(1,2),(1,3)\dots ..(1,6\left)\right(2\mid ,1)\dots .(6,6\left)\right\}$

Let $X$ denote the sum of the numbers on the
two dile then the range of $x=\{2,3,4\dots .12\}$

Mean $X=\mu =\sum _{i=2}^{12} xiP(x=xi)=2\cdot \frac{1}{36}+3\cdot \frac{2}{36}+\dots \dots +12\cdot \frac{1}{36}$

$=\frac{1}{36}[2+6+12+20+30+42+40+36+30+22+12]=\frac{252}{36}=7$