firstly, for a) is the generating function (s(1-p) + pF(s))^k?
and for b) do you assume that first some individuals have offspring, then the offspring each mature with probability p, then you count the number of mature individuals
#probability generating functions
warm smelt
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firstly, for a) is the generating function (s(1-p) + pF(s))^k?
and for b) do you...
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warm smelt
my reasoning for a) is like
if X_1, ... X_k are random vars where X_i is the number of immature individuals created by immature individual i, then we want (P(s))^k because they are iid
(P is pgf for X_i)
and if X_i stays immature then this corresponds to a term s(1-p)
if X_i matures then the pgf is F(s) so this gives a term pF(s)
in total (s(1-p)+pF(s))^k, and the derivative is k(s(1-p)+pF(s))^{k-1}(F'(s)+1-p) and then just evaluate at 1 to get mean
But wait are you supposed to assume for both parts that some of the offspring mature before you count
😔
actually i assume you do
in that case for b) let Y_i = 1+the number of mature children of individual i. the generating function is Q(s)^k where Q is the pgf for Y_i. from some expansion I get this equal to F(1-p)^k[z/(1 - zp/(1-p))]^k
but from the position of the question on the sheet i think it should be simpler than this
obsidian ice
and if X_i stays immature then this corresponds to a term s(1-p)
I don't think we're talking about organisms that mature or "stay immature". I think the probability of maturation is a probability of surviving to maturity.
I think a is just asking how many offspring are born into generation 2.
warm smelt
OH my bad that makes sense