Question 1 Let X be a continuous random variable taking values in [a, b] with c.d.f. Fx which is strictly increasing on [a, b]. (a) Show that the random variable Fx (X) has a uniform distribution on [0, 1]. (b) Let U be a uniform random variable on [0, 1]. What is the distribution of the random variable F¹(U), where Fx¹ is the inverse of Fx? X (c) Suppose that U₁, U2,..., Un are a set of computer-generated pseudo-random numbers (assumed to be drawn from a uniform distribution on [0, 1]). How would you use them to simulate a random sample X₁, X2,..., Xn from the distribution with density f(x) = μe-μα, x ≥ 0?

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