Pascua see polynomial function g(z) is defined like that….
$g_{x}(z)=\sum_{0}^{N}p_{j}z^{^{j}}$ for P(X=j)=$p_{j}$, j∈{0,…,N}
Meaning-
suppose variable x has probability p0,p1,p2..at x=0 x=1 and x=2 respectively then
$g_{x}(z)=p_{0}+p_{1}z+p_{2}z^{2}+...$ by definition of given function….
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now we are given that—
$g{_{y}}$(z)=$(1−β+βz)^{^{N}}$ and we assume N=1
then
$g{_{y}}$(z)=1−β+βz
so
coefficient of $z^{^{0}}$ gives probability at y=0 i.e p0 and
coefficient of z gives probability at y=1 i.e p1 .
Expectation of Y is β at N=1 ..only option B matching…..
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We can verify by taking any value eg. N=2