91Ó°ÊÓ

Use the rejection method with g(x) = 1, 0 < x < 1, to determine an algorithm for simulating a random variable having density function

$f\left(x\right)\left\{\begin{array}{ll}60x^3{(1-x)}^2& 0<x<1\\ 0& otherwise\end{array}\right.$

Let (X, Y) be uniformly distributed in the circle of radius 1 centered at the origin. Its joint density is thus

$f(x,y)=\frac{1}{\mathrm{π}}0≤x^2+y^2≤1$

Let R = (X2 + Y2)1/2 and = tan−1(Y/X) denote

the polar coordinates of (X, Y). Show that R and are

independent, with R2 being uniform on (0, 1) and being

uniform on (0, 2Ï€).

Suppose we have a method for simulating random variables from the distributions F1 and F2. Explain how to simulate from the distribution

F(x) = pF1(x) + (1 − p)F2(x) 0 < p < 1

Give a method for simulating from

$F\left(x\right)\left\{\begin{array}{ll}\frac{1}{3}(1-e^{-3x})+\frac{2}{3}x& 0<x≤1\\ \frac{1}{3}(1-e^{-3x})+\frac{2}{3}& x>1\end{array}\right.$