Question

If x1,x2,x3 are independent random variables that are uniformly distributed over (0, 1). show that the probability that the largest of the three is greater than the sum of the other two is

a) 0
b) 0.25
c) 0.5
d) 1

Answer 1

The probability that the largest of three independent random variables is greater than the sum of the other two is 0.5.

Step-by-step explanation:

To find the probability that the largest of three independent random variables is greater than the sum of the other two, we need to determine the range of values for which this is true.

Let's assume that the largest variable is x1, and the other two are x2 and x3.

In order for x1 to be greater than the sum of x2 and x3, the sum of x2 and x3 must be less than x1. Therefore, we need to find the probability that x2 + x3 is less than x1.

Since x1, x2, and x3 are uniformly distributed over (0, 1), we can represent their distributions as continuous probability functions. The probability of x2 + x3 being less than x1 is equal to the area under the probability distribution curve of x1, to the left of any given point. This area is 0.5, or option c) 0.5.