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Chapter 1: Problem 6

If the sample space is \(\mathcal{C}=\\{c:-\infty

Short Answer

Expert verified
The given set function is not a probability set function as its integral over the sample space is 2, not 1. By multiplying it by \(1/2\), it becomes a probability set function.

Step by step solution

01

Identify the Function

The function given is \(f(x) = e^{-|x|}\). This function must integrate to 1 over the sample space \(\mathcal{C}=\{c:-\infty<c<\infty\}\) to constitute a probability set function.
02

Check Integration over Given Sample Space

Firstly, integrate the given function over the sample space. Note that due to symmetry, it could be calculated for half of the space and doubled. So, \(\int_{-\infty}^{\infty} e^{-|x|} dx = 2 \int_{0}^{\infty} e^{-x} dx\). Simple integration gives \(2[-e^{-x}]_{0}^{\infty}\) which results in 2.
03

Analyze the Result

Since \(f(x)\) integrates to 2 over the sample space, and not 1, it's not a probability set function. A probability set function's integral over the sample space must be equal to 1.
04

Adjust to Make It a Probability Set Function

To make \(f(x)\) a probability set function, it must be multiplied by a constant such that it integrates to 1 over the sample space. If \(f(x)\) integrates to 2, then the function would integrate to 1 if we divided it by 2. Therefore, the constant to multiply the integrand by to make it a probability set function is \(1/2\).

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