Probability | Part IA, 2004

Let XX be a positive-integer valued random variable. Define its probability generating function pXp_{X}. Show that if XX and YY are independent positive-integer valued random variables, then pX+Y=pXpYp_{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(2,2,2,3,5,7 ) and (1,1,5,6,7,8))(1,1,5,6,7,8)). Show that there exists a non-standard pair of dice AA and BB such that when thrown

P{P\{ total shown by AA and BB is n}=P{n\}=P\{ total shown by pair of ordinary dice is n}n\}

for all 2n122 \leqslant n \leqslant 12.

[Hint: (x+x2+x3+x4+x5+x6)=x(1+x)(1+x2+x4)=x(1+x+x2)(1+x3).]\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]

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