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Chapter 1: Problem 6

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

Short Answer

Expert verified
This set function is not a probability set function because the integral over the function across the entire space equals 2, not 1. The constant to make \( \int_{C} e^{-|x|} dx \) a valid probability set function is \( \frac{1}{2} \).

Step by step solution

01

Define the set function

We define the set function \( f(C) \) to be \( \int_{C} e^{-|x|} dx \). This function gives us the area underneath the curve \( e^{-|x|} \) from \( -\infty \) to \( \infty \).
02

Evaluate the integral on the entire space

Evaluate \( f(\mathcal{C}) \), where \( \mathcal{C} = \{-\infty<c<\infty\} \), by integrating \( e^{-|x|} \) from \( -\infty \) to \( \infty \). This integral can be broken down into two parts due to absolute value, \( \int_{-\infty}^{0} e^{x} dx + \int_{0}^{\infty} e^{-x} dx \), which evaluates to 2, not 1.
03

Determine if the set function is a probability set function

Since \( f(\mathcal{C}) = 2 \) and not 1 as required by the definition of a probability set function, we see that \( f \) alone can't serve as a probability set function.
04

Find the constant to make it a probability set function

Since we need the integral over the entire space to be 1 for a probability set function, we must scale down the original function by a factor of 2 to make it a probability set function. Thus, the constant is \( \frac{1}{2} \).

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