(a) Let W be a random variable giving the number of heads minus the number of tails in four tosses of a coin.

(i) Find the probability distribution of the random variable W, assuming that the coin is biased so that a head is twice as likely to occur as a tail.

(ii) Calculate the expected value of W.

(b) Measurements of scientific systems are always subject to variation, some more than others. There are many structures for measurement error. Suppose the measurement error X of a certain physical quantity is decided by the density function:

() = (3 2)-1 x 1

= 0 elsewhere

(i) Determine k that renders f(x) a valid density function.

(ii) Find the probability that a random error in measurement is less than 1/4.

(iii) Can a probability histogram be constructed for f(x) .Give reason for your answer?