A defective coin minting machine produces coins whose probability of heads is a random variable P with PDF

f_P(p)=pe^p , p=[0, 1]. and 0, otherwise.

A coin produced by this machine is selected and tossed repeatedly, with successive tosses assumed independent.

a) Find the probability that a coin toss results in heads.

b) Given that a coin toss resulted in heads, find the conditional PDF of P.

c) Given that the first coin toss resulted in heads, find the conditional probability of heads on the next toss.