Question

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

Short answer

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

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 $-\mathrm{\infty }$ to $\mathrm{\infty }$.

Evaluate the integral on the entire space

Evaluate $f(\mathcal{C})$, where $\mathcal{C}=\{-\mathrm{\infty }<c<\mathrm{\infty }\}$, by integrating $e^{-|x|}$ from $-\mathrm{\infty }$ to $\mathrm{\infty }$. This integral can be broken down into two parts due to absolute value, ${\int }_{-\mathrm{\infty }}^0{e}^{x}dx+{\int }_0^{\mathrm{\infty }}{e}^{-x}dx$, which evaluates to 2, not 1.

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.

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 $\frac{1}{2}$.