1. Suppose that random variable X, representing the proportion of Fred's budget de- voted to transport, has a probability density function (pd. f.) given by al cm: 0 S cc 3 l f(:v|a) _ { 0 otherwise, where parameter a > 0. (a) Derive an expression for the probability that Fred spends between half and threequarters of his budget on transport. Evaluate this numerically for the case where a: = 2. (b) Show that the kth moment for X is a/(k + a), where k is a positive integer. Hint: p.d.f.s integrate to unity. (c) Hence derive an expressiOn for the variance of X. ((1) Compute the value of E[(X+\"))3] for the case where a = 2. (e) What does this numerical result imply about the shape of the p.d.f.? Plot (using software of your choice, such as Python) the p.d.f. to support your conclusion