Page 201 - Probability and Statistical Inference

178 4. Functions of Random Variables and Sampling Distribution

problem using the direct calculation, and then provide an alternative quick
and painless way to solve the same problem by applying a result derived later
in this chapter.
Example 4.1.1 Let us consider a popular game of “hitting the bull’s eye” at
the point of intersection of the horizontal and vertical axes. From a fixed
distance, one aims a dart at the center and with the motion of the wrist, lets it
land on the game board, say at the point with the rectangular coordinates (Z ,
1
Z ). Naturally then, the distance (from the origin) of the point on the game
2
board where the dart lands is A smaller distance would indicate
better performance of the player. Suppose that Z and Z are two indepen-
2
1
dent random variables both distributed as N(0, 1). We wish to calculate
where a > 0, that is the probability that the thrown dart
lands within a distance of from the aimed target. The joint pdf of (Z , Z )
1
2
would obviously amount to )} with
–∞ < z , z < ∞. We then go back to the heart of (3.3.20) as we proceed to
1
2
evaluate the double integral,
Let us substitute

which transform a point (z , z ) on the plane in the rectangular coordinates
1
2
system to the point (r, θ) on the same plane, but in the polar coordinates
system. Now, the matrix of the first partial derivatives of z and z with re-
1
2
spect to θ and r would be

so that its determinant, det(A) = ½. Now, we can rewrite the double integral
from (4.1.1) as

But, later in this chapter (Example 4.3.6), we show that the distribution
of is indeed , that is a Chi-square with two degrees of