Question

Assess a subjective triangular probability distribution for the random variable, X, where X is defined as the amount of snowfall you think we will get in our next snowstorm in inches. Suppose that you expect your commute to increase in minutes according to the function $f\left(x\right)=0.2x^2$ . What is the expected increase in your commute for the next snowstorm?

Estimate $E\left[f\left(X\right)\right]$ E[f(X)] by creating simple Monte Carlo simulation in Excel.

Here are some equations for the triangular distribution that may help you:

a is the lower limit, b is the upper limit, and c is the mode

PDF: $f\left(x\right)=\begin{cases} 0 \qquad \qquad \qquad \text{ for } x<a\\ \frac{2\left(x-a\right)}{\left(b-a\right)\left(c-a\right)} \text{ for } a \le x \le c\\ \frac{2\left(b-x\right)}{\left(b-a\right)\left(b-c\right)} \text{ for } c<x \le b\\ 0 \qquad \qquad \qquad \text{ for } x>b \end{cases}$

CDF: $F\left(x\right)=\begin{cases} 0 \qquad \qquad \qquad \qquad \text{ for } x<a\\ \frac{\left(x-a\right)^2}{\left(b-a\right)\left(c-a\right)} \qquad \text{ for } a \le x \le c\\ 1-\frac{\left(b-x\right)^2}{\left(b-a\right)\left(b-c\right)} \; \text{ for } c<x \le b\\ 1 \qquad \qquad \qquad \qquad \text{ for } x>b \end{cases}$

Inverse CDF: $F^{-1}\left(x\right)=\begin{cases} \sqrt{\left(b-a\right)\left(c-a\right)x}+a \qquad \quad \; \text{ for } 0\le x\le\frac{c-a}{b-a} \\ \:b-\sqrt{\left(b-a\right)\left(b-c\right)\left(1-x\right)} \quad \text{ for } \frac{c-a}{b-a}<x\le1 \end{cases}$

Answer 1

the random variable X, the amount of snowfall has a triangular distribution with the following cdf

We also know the inverse CDF

$F^{-1}(x)=\begin{cases}\sqrt{(b-a)(c-a)x}+a,&0\le x\le \frac{c-a}{b-a}\\ b-\sqrt{(b-a)(b-c)(1-x)},&\frac{c-a}{b-a}<x\le 1 \end{cases}$

We will use the inverse cdf method to generate X, using the following steps

1. Draw a number u from uniform (0,1)
2. $x=F^{-1}(u)$ is a draw from X
   • where $F^{-1}(u)=\begin{cases}\sqrt{(b-a)(c-a)u}+a,&0\le u\le \frac{c-a}{b-a}\\ b-\sqrt{(b-a)(b-c)(1-u)},&\frac{c-a}{b-a}<u\le 1 \end{cases}$

We will set up the following excel

Duplicate trial 1 to 100 rows

get the following (paste the random number as values to avoid regenartion of numbers)

The expectation of f(x) is the average of f(x) that we calculated in column D

Get the following

the expected increase in your commute for the next snowstorm (in this realization of 100 trials) is

E[f(x)] = 5.81 minutes.

The accuracy of this estimate would improve as you increase the number of trials from 100 to say 1000