Question

Q3. The aim of this question is to see a typical use-case for the linearity of expectation. Consider an experiment in which we toss a biased coin (probability of heads = p) n times. Let Y be the random variable that is the number of heads. Also, let Xi be the 0/1 random variable that is 1 if the ith toss was heads and 0 otherwise.

(b) [5 pts] Let 1 ≤ i ≤ n be any integer. What is E[Xi ]?

(c) [15 pts] Find a closed form expression for E[Y ] using part (b). [Hint: observe the relationship between Y and the Xi and use the linearity of expectation.]

Answer 1

given that

P(head)=p

Y is number of heads out of "n" tosses

Xi =1 if toss result in head

=0 if toss result in tail

b)

E(Xi) =1*P(Xi=1) +0*P(Xi=0)

=1*P(head) +0*P(tail)

=1*p +0*(1-p) =p

c)

since n is total number of tosses

while Y is number of heads in n tosses

Xi takes value 1 if toss results in head otherwise its zero hence

Y=X1+X2+X3+.....+Xn

so now