Let X be a exponential random variable with pdf f(x) = ?e??x forx > 0, and cumulative distribution function F(x).
(a) Show that F(x) = 1?e ??x for x > 0, and show that thisfunction satisfies the requirements of a cdf (state what these are,and show that they are met). [4 marks]
(b) Draw f(x) and F(x) in separate graphs. Define, and identifyF(x) in the graph of f(x), and vice versa. [Hint: write themathematical relationships, and show graphically what the functionsrepresent.] [4 marks]
(c) X has mgf M(t) = ?(??t) ?1 . Derive the mean of the randomvariable from first principles (i.e. using the pdf and thedefinition of expectation). Also show how this mean can be obtainedfrom the moment generating function. [10 marks]
(d)
(i) Show that F ?1 (x) = ? 1 ? ln(1 ? x) for 0 < x < 1,where ln(x) is the natural logarithm. [4 marks]
(ii) If 0 < p < 1, solve F(xp) = p for xp, and explainwhat xp represents. [4 marks] (iii) If U ? U(0, 1) is a uniformrandom variable with cdf FU (x) = x (for 0 < x < 1), provethat X = ? 1 ? ln(1 ? U) is exponential with parameter ?. Hence,describe how observations of X can be simulated. [4 marks]
