Problem 1.73 A laboratory keeps a population of aphids. The probability of an aphid passing a day uneventfully is q

Extracted text: Problem 1.73 A laboratory keeps a population of aphids. The probability of an aphid passing a day uneventfully is q< 1.="" given="" that="" a="" day="" is="" not="" uneventful,="" there="" is="" a="" probability="" r="" that="" the="" aphid="" will="" have="" one="" offspring,="" a="" probability="" s="" that="" it="" will="" have="" two="" offspring,="" and="" a="" probability="" t="" that="" it="" will="" die="" of="" exhaustion,="" where="" r+s+t="1." offspring="" are="" ready="" to="" reproduce="" the="" next="" day.="" the="" fates="" of="" different="" aphids="" are="" independent,="" as="" are="" the="" events="" of="" different="" days.="" the="" laboratory="" starts="" out="" with="" one="" aphid.="" let="" x,="" be="" the="" number="" of="" aphids="" at="" the="" end="" of="" n="" days.="" show="" how="" to="" obtain="" an="" expression="" for="" the="" pgf="" d(z)="" of="" x„.="" what="" is="" the="" expected="" value="" of="" x,?="" show="" that="" the="" probability="" of="" extinction="" does="" not="" depend="" on="" q="" and="" that="" if="" 2r="" +="" 3s=""><1, the="" aphids="" will="" certainly="" die="" out.="" find="" the="" probability="" of="" extinction="" if="" r="1/5," s="2/5" and="" 1="">