Provide correct answers.

18.10 Transforming a random variable Find the mean and variance of the continuous random variable X with probability density function given by ( 3x 4. 121, fx (x) = otherwise. A new random variable 1 is defined by the relation Y = X . Find the cumulative distribution function of Y, and hence derive its probability density function. Verify that, for X and Y as given above, E(XY ) # E(X ) E(Y), and explain why this result holds in this particular example.18.8 Finding the mode and median of a distribution A continuous random variable X has probability density function /x(x ) given by fx(x ) = k(2 -x)(x - 5), 25x55, = 0. elsewhere. Find the value of & , and hence deduce the mean and variance of X, What are the values of the mode and median of the distribution of X? SolutionIB.5 The randomised response experiment Let 0 denote the probability that a randomly sampled individual in some population voted Conservative in the last General Election. In a particular type of 'randomised response experiment, cach of a random sample of individuals from this population responds 'True' or 'False' to one of the following two statements. (a) I voted Conservative at the last General Election. (b) I did not vote Conservative at the last General Election. A randomising device is used to ensure that the probabilities of responding to (a) and (b) are p and 1 -p respectively, where p is known and 0