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Chapter 10: Q. 10.5 (page 431)
Use the inverse transformation method to present an approach for generating a random variable from the Weibull distribution
The required method is used for the solution and it is explained below.
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In Example 4a, we showed that
E[(1 − V2) 1/2] = E[(1 − U2) 1/2] = π/4
when V is uniform (−1, 1) and U is uniform (0, 1). Now show that
Var[(1 − V2) 1/2] = Var[(1 − U2) 1/2]
and find their common value.
Develop a technique for simulating a random variable having density function
Give a method for simulating a random variable having failure rate function (a) λ(t) = c;
(b) λ(t) = ct;
(c) λ(t) = ct2;
(d) λ(t) = ct3.
Present a method for simulating a random variable having distribution 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.
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