Question

Let X be a uniform random variable with pdf f(x) = λe−λx for x > 0, and cumulative distribution function F(x).

(a) Show that F(x) = 1−e −λx for x > 0, and show that this function 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 identify F(x) in the graph of f(x), and vice versa. [Hint: write the mathematical relationships, and show graphically what the functions represent.] [4 marks]

(c) X has mgf M(t) = λ(λ−t) −1 . Derive the mean of the random variable from first principles (i.e. using the pdf and the definition of expectation). Also show how this mean can be obtained from the moment generating function. [10 marks]

(d)

(i) Show that F −1 (x) = − 1 λ 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 explain what xp represents. [4 marks] (iii) If U ∼ U(0, 1) is a uniform random variable with cdf FU (x) = x (for 0 < x < 1), prove that X = − 1 λ ln(1 − U) is exponential with parameter λ. Hence, describe how observations of X can be simulated. [4 marks]

Answer 1