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Chapter 10: Q. 10.12 (page 431)

Explain how you could use random numbers to approximate ∫01k(x)dx, where k(x) is an arbitrary function.

Short Answer

Expert verified

The answer of the question is∫01k(u)du=E(k(U))=1n∑i=1nk(Ui)

Step by step solution

01

Given Information

We need to explain the use of random numbers to approximate∫01k(x)dx

02

Simplify

Consider U~Unif(0,1).Using the theorem about the expectation of the function of the random variable, we have that

E(k(u))=∫ℝk(u)fU(u)du=∫01k(u)du

On the other hand, we can estimate the mean of random variable with the mean of the sample. So, taking the independent variable U1,...,Un~Uand considering its functional values localid="1648210961949" k(U1),...,k(Un).Therefore, we can estimate

E(k(U))=1n∑k(Uii=1n)

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Most popular questions from this chapter

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.

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) … x} = xn

(c) Use part (b) to give a second method of simulating a random variable having distribution F.

In Example 2c we simulated the absolute value of a unit normal by using the rejection procedure on exponential random variables with rate 1. This raises the question of whether we could obtain a more efficient algorithm by using a different exponential density—that is, we could use the density g(x) = λe−λx. Show that the mean number of iterations needed in the rejection scheme is minimized when λ = 1.

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.

Let X and Y be independent exponential random variables with mean 1.

(a) Explain how we could use simulation to estimate E[eXY].

(b) Show how to improve the estimation approach in part (a) by using a control variate.
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