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Chapter 10: Q. 10.4 (page 430)
Present a method for simulating a random variable having distribution function
The universality of uniform is used to obtain the required method.
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If X is a normal random variable with mean μ and variance σ2, define a random variable Y that has the same distribution as X and is negatively correlated with it.
Give a technique for simulating a random variable having the probability density function
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—that is, when the value of P(k) is initially set—P(1),P(2), ... ,P(k) is a random permutation of 1, 2, ... , k.
Suppose it is relatively easy to simulate from Fi for each i = 1, ... , n. How can we simulate from
(a)
(b)
Explain how you could use random numbers to approximate , where k(x) is an arbitrary function.
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