Introduction to Probability and Statistics for Engineers and Scientists

166 Chapter 5: Special Random Variables

U 1 < .4 U 1 > .4

U 2 < .25 U 2 > .25 U 2 < .5 U 2 > .5

U 3 < –^13 U 3 > –^13 U 3 < –^13 U 3 > –^13 U 3 < –^23

U 4 < –^12 U 4 > –^12 U 4 > –^12 U 4 < –^12 U

4 –^12

U 4 > –^12

S = {1, 3}

S = {1, 2}

S = {2, 3}

U 3 > –^23

S = {4, 5}

S = {1, 4} S = {1, 5} S = {2, 4} S = {2, 5}S = {3, 4} S = {3, 5}

FIGURE 5.6 Tree diagram.

This process stops whenI 1 +···+Ij=kand the random subset consists of thekelements
whoseI-value equals 1. That is,S={i:Ii= 1 }is the subset.
For instance, ifk =2,n= 5, then the tree diagram of Figure 5.6 illustrates the
foregoing technique. The random subsetSis given by the final position on the tree. Note
that the probability of ending up in any given final position is equal to 1/10, which can be
seen by multiplying the probabilities of moving through the tree to the desired endpoint.
For instance, the probability of ending at the point labeledS ={2, 4}is P{U 1 >
.4}P{U 2 <.5}P{U 3 >^13 }P{U 4 >^12 }=(.6)(.5)

( 2

3

)( 1

2

)
=.1.
As indicated in the tree diagram (see the rightmost branches that result inS={4, 5}),
we can stop generating random numbers when the number of remaining places in the
subset to be chosen is equal to the remaining number of elements. That is, the general

procedure would stop whenever either

∑j

i= 1 Ii=kor

∑j
i= 1 Ii=k−(n−j). In the
latter case,S={i≤j:Ii=1,j+1,...,n}. ■

EXAMPLE 5.4e The random vectorX,Y is said to have auniformdistribution over the
two-dimensional regionRif its joint density function is constant for points inR, and is 0
for points outside ofR. That is, if

f(x,y)=

{

c if (x,y)∈R

0 if otherwise