Question

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

Short answer

\(P\left( {{X^2} + {Y^2} + {Z^2} \le 1} \right) < 0.2\)

Step-by-step solution

Given information

X, Y, and Z are independent random variables, and each of the variables follows the standard normal distribution.

Calculate the probability

Since \({X^2} + {Y^2} + {Z^2}\)has \({\chi ^2}\)distributed with degrees of freedom 3.

Referring to the table at the end of the book for the answer.

Hence, \(P\left( {{X^2} + {Y^2} + {Z^2} \le 1} \right) < 0.2\).