[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

In Exercise Exercise,

the distribution came “out of a hat.” In this problem, we will again consider an experiment whose outcomes are not equally likely. We will determine a function [math]f(x)[/math] which can be used to determine the probability of certain events. Let [math]T[/math] be the right triangle in the plane with vertices at the points [math](0, 0),\ (1, 0),[/math] and [math](0,1)[/math]. The experiment consists of picking a point at random in the interior of [math]T[/math], and recording only the [math]x[/math]-coordinate of the point. Thus, the sample space is the set [math][0,1][/math], but the outcomes do not seem to be equally likely. We can simulate this experiment by asking a computer to return two random real numbers in [math][0, 1][/math], and recording the first of these two numbers if their sum is less than 1. Write this program and run it for 10,00 trials. Then make a bar graph of the result, breaking the interval [math][0, 1][/math] into 10 intervals. Compare the bar graph with the function [math]f(x) = 2 - 2 x[/math]. Now show that there is a constant [math]c[/math] such that the height of [math]T[/math] at the [math]x[/math]-coordinate value [math]x[/math] is [math]c[/math] times [math]f(x)[/math] for every [math]x[/math] in [math][0, 1][/math]. Finally, show that

[[math]] \int_0^1 f(x)\,dx = 1\ . [[/math]]

How might one use the function [math]f(x)[/math] to determine the probability that the outcome is between [math].2[/math] and [math].5[/math]?