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Q. 10.16

Expert-verified

A First Course in Probability

Found in: Page 431

Book edition 9th

Author(s) Sheldon M. Ross

Pages 432 pages

ISBN 9780321794772

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Short Answer

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 required statement is proved below.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information

We need to prove the statement.

Step 2: Explanation

Consider $\frac{g (X)}{f (X)}$ is a random variable, as the function of a random variable $X$ . Using the theorem regarding the expectation of the function of the random variable, we have

$E [\frac{g (X)}{f (X)}] = \int_{0}^{1} \frac{g (x)}{f (x)} \cdot f (x) d x = \int_{0}^{1} g (x) d x$

Since, we have the expected value of $\frac{g (X)}{f (X)}$ is equal to the required value $\int_{0}^{1} g (x) d x$ , we have that it is a good estimater of $\frac{g (x)}{f (x)}$ . In statistics, these estimators that have the mean equal to the estimated value are called unbiased estimators.

Most popular questions for Math Textbooks

Let (X, Y) be uniformly distributed in the circle of radius 1 centered at the origin. Its joint density is thus

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(a) Give a method for simulating a random variable having distribution F that uses only a single random number.

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

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$F (x) = \{\begin{cases} 0 & x \leq - 3 \\ \frac{1}{2} + \frac{x}{6} & - 3 < x < 0 \\ \frac{1}{2} + \frac{x^{2}}{32} & 0 < x \leq 4 \\ 1 & x > 4 \end{cases}$

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