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please send me answer in type form strictly prohibited handwritten solution and don't use AI show your own work with step by step explanation.

Consider a population of animals in which each mature individual produces a random number of offspring with generating function F. Suppose we start with a population of k immature individuals , each of which grows to maturity with probability p , independently of the other individuals .

( a ) Find the generating function for the distribution of the number of immature individuals in the next generation .

( b ) Find the generating function for the distribution of the number of mature individuals in the next generation , given that there are k mature individuals in the parent generation .

( c ) Show that the distributions in ( a ) and ( b ) have the same mean , but not necessarily the same variance .

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