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Chapter 10: Q. 10.2 (page 431)
Give an approach for simulating a random variable having probability density function
f(x) = 30(x2 鈭� 2x3 + x4) 0 < x < 1
The method for simulating a random variable is the rejection method.
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The following algorithm will generate a random permutation of the elements 1, 2, ... , n. It is somewhat faster than the one presented in Example 1a but is such that no position is fixed until the algorithm ends. In this algorithm, P(i) can be interpreted as the element in position i. Step 1. Set k = 1. Step 2. Set P(1) = 1. Step 3. If k = n, stop. Otherwise, let k = k + 1. Step 4. Generate a random number U and let P(k) = P([kU] + 1) P([kU] + 1) = k Go to step 3. (a) Explain in words what the algorithm is doing. (b) Show that at iteration k鈥攖hat is, when the value of P(k) is initially set鈥擯(1),P(2), ... ,P(k) is a random permutation of 1, 2, ... , k.
Give a technique for simulating a random variable having the probability density function
Let (X, Y) be uniformly distributed in the circle of radius 1 centered at the origin. Its joint density is thus
Let R = (X2 + Y2)1/2 and = tan鈭�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, 2蟺).
Develop a technique for simulating a random variable having density function
Suppose it is relatively easy to simulate from Fi for each i = 1, ... , n. How can we simulate from
(a)
(b)
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