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Chapter 10: Q. 10.3 (page 430)
Give a technique for simulating a random variable having the probability density function
The technique for simulating a random variable is the universality of unform.
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In Example 4a, we showed that
E[(1 鈭� V2) 1/2] = E[(1 鈭� U2) 1/2] = 蟺/4
when V is uniform (鈭�1, 1) and U is uniform (0, 1). Now show that
Var[(1 鈭� V2) 1/2] = Var[(1 鈭� U2) 1/2]
and find their common value.
Let F be the distribution function
F(x) = xn 0 < x < 1
(a) Give a method for simulating a random variable having distribution F that uses only a single random number.
(b) Let U1, ... , Un be independent random numbers. Show that
P{max(U1, ... , Un) 鈥� x} = xn
(c) Use part (b) to give a second method of simulating a random variable having distribution F.
Give an approach for simulating a random variable having probability density function
f(x) = 30(x2 鈭� 2x3 + x4) 0 < x < 1
Let X be a random variable on (0, 1) whose density is f(x). Show that we can estimate # 1 0 g(x) dx by simulating X and then taking g(X)/f(X) as our estimate. This method, called importance sampling, tries to choose f similar in shape to g, so that g(X)/f(X) has a small variance.
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.
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