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Q. 10.16

A First Course in Probability
Found in: Page 431
A First Course in Probability

A First Course in Probability

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.

See the step by step solution

Step by Step Solution

Step 1: Given Information

We need to prove the statement.

Step 2: Explanation

Considerg(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


Since, we have the expected value of g(X)f(X) is equal to the required value 01g(x)dx, we have that it is a good estimater of g(x)f(x). In statistics, these estimators that have the mean equal to the estimated value are called unbiased estimators.

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