5. Suppose that a point (X, Y, Z) is to be chosen at random in three-dimensional space, where X, Y, and Z are independent random variables and each has the standard normal distribution. What is the probability that the distance from the (0,0,0) to the point will be less than 1?

5. Suppose that a point (X, Y, Z) is to be chosen at random in three-dimensional space, where X, Y, and Z are independent random variables and each has the standard normal distribution. What is the probability that the distance from the (0,0,0) to the point will be less than 1?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 27T

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Question

5. Suppose that a point (X, Y, Z) is to be chosen at random in three-dimensional space,
where X, Y, and Z are independent random variables and each has the standard normal
distribution. What is the probability that the distance from the (0,0,0) to the point will
be less than 1?

Expert Solution

Step 1: Given information.

Given that:

$open parentheses X comma Y comma Z close parentheses$ is a point chosen in three-dimensional space.
$X comma space Y comma space and space Z$ are three independent standard normal random variables.

To find:

The probability that the distance from the (0,0,0) to the point will be less than 1.

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