Xi takes either 1(with prob p) or 0(with prob 1-p ).

Means Xi is nothing but Bernoulli Distribution. so the pdf of an Xi can be written as

f(xi)=p , xi=1

=1-p , xi=0

∑Xi= 100 implies any 100 Xi's out of total n Xi's should be 1.

f(y) = probability of any (exactly) 100 Xi's out of total n Xi's to be 1 ; y=1 (in such a case)

= 1- probability of any(exactly) 100 Xi's out of total n Xi's to be 1 ; y=0 (in such a case)

ie Y~bin(n,p)

So,

f(y)= nC100* p ^100 * (1-p)^(n-100) ; y=1

=1- nC100* p ^100 * (1-p)^(n-100) ; y=0

E(Y^2)= 1^2* nC100* p ^100 * (1-p)^(n-100) + 0^2 * [1- nC100* p ^100 * (1-p)^(n-100)]

= nC100* p ^100 * (1-p)^(n-100)

option

**b**
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"You don't have to believe in God, but you should believe in The Book." -Paul Erdős