Provide step by step solution to the following questions.

The random variable X has a normal distribution with mean # and variance o', such that the probability density function of X is: fx (x) (i) Determine the probability density function of the random variable Y, where Y = 2X . [3] (ii) Hence, show that Y is normally distributed with mean 2/ and variance 40-. [!] (iii) Assuming the moment generating function of a N(#,0 ) random variable, prove the result obtained in part (ii). [3] [Total 7]At the end of each year an independent organisation ranks the performance of the unit trusts invested in a particular sector, and classifies them into four quartiles (with quartile I being the best performer). Past experience has shown that, at the end of each year, a fund will either remain in the same quartile or will move to a neighbouring quartile. In fact, there is a probability 1-2c that a fund will remain in the same quartile and, where upward or downward movements are both possible, these are equally likely. However, it has been found that a fund that has remained in the top or bottom quartile for two consecutive years has a probability of 1- (