# Probability that the Largest of Three Independent Uniformly Distributed Random Variables Exceeds the Sum of the Other Two

I am currently studying probability and statistics and I came across a problem that I am having trouble with. I have some understanding of random variables and their distributions, but this particular problem has left me stumped.

Problem Statement:

Let $$X_1, X_2, X_3$$ be independent random variables that are uniformly distributed over (0,1). I am trying to compute the probability that the largest of the three is greater than the sum of the other two.

My Attempt:

I understand that the probability density function (pdf) of a uniformly distributed random variable over the interval (0,1) is given by:

$$f(x) = \begin{cases} 1 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$$

Since the random variables are independent, I know that the joint pdf is the product of the individual pdfs. However, I am not sure how to proceed from here to find the desired probability.

Background:

I am an undergraduate student with a basic understanding of probability, statistics, and calculus. I have studied random variables and their distributions, and I am familiar with the concepts of pdf and cumulative distribution function (cdf).

Motivation:

This problem is part of a set of exercises in my course on probability and statistics. Understanding how to solve this problem will help me better understand the behavior of uniformly distributed random variables and the concept of independence.

I would greatly appreciate any help or hints on how to approach this problem. Thank you in advance!

• Try thinking about it geometrically -- the three values are uniformly distributed over the unit cube. What does the region where $z>x+y$ look like? What is its volume? Then apply symmetry for $x>y+z$ and $y>x+z$ May 15 at 12:27