Let $X$ be a positive-integer valued random variable. Define its probability generating function $p_{X}$. Show that if $X$ and $Y$ are independent positive-integer valued random variables, then $p_{X+Y}=p_{X} p_{Y}$.

A non-standard pair of dice is a pair of six-sided unbiased dice whose faces are numbered with strictly positive integers in a non-standard way (for example, $(2,2,2,3,5,7$ ) and $(1,1,5,6,7,8))$. Show that there exists a non-standard pair of dice $A$ and $B$ such that when thrown

$P\{$ total shown by $A$ and $B$ is $n\}=P\{$ total shown by pair of ordinary dice is $n\}$

for all $2 \leqslant n \leqslant 12$.

[Hint: $\left.\left(x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}\right)=x(1+x)\left(1+x^{2}+x^{4}\right)=x\left(1+x+x^{2}\right)\left(1+x^{3}\right) .\right]$