Introduction to Probability and Statistics for Engineers and Scientists

104 Chapter 4:Random Variables and Expectation

In particular, for anynsets of real numbersA 1 ,A 2 ,...,An

P{X 1 ∈A 1 ,X 2 ∈A 2 ,...,Xn∈An}

=

∫

An

∫

An− 1

...

∫

A 1

f(x 1 ,...,xn)dx 1 dx 2 ...dxn

The concept of independence may, of course, also be defined for more than two random
variables. In general, thenrandom variablesX 1 ,X 2 ,...,Xnare said to be independent if,
for all sets of real numbersA 1 ,A 2 ,...,An,

P{X 1 ∈A 1 ,X 2 ∈A 2 ,...,Xn∈An}=

∏n

i= 1

P{Xi∈Ai}

As before, it can be shown that this condition is equivalent to

P{X 1 ≤a 1 ,X 2 ≤a 2 ,...,Xn≤an}

=

∏n

i= 1

P{X 1 ≤ai} for alla 1 ,a 2 ,...,an

Finally, we say that an infinite collection of random variables is independent if every finite
subcollection of them is independent.

EXAMPLE 4.3e Suppose that the successive daily changes of the price of a given stock are
assumed to be independent and identically distributed random variables with probability
mass function given by

P{daily change isi}=























−3 with probability .05

−2 with probability .10

−1 with probability .20

0 with probability .30

1 with probability .20

2 with probability .10

3 with probability .05

Then the probability that the stock’s price will increase successively by 1, 2, and 0 points
in the next three days is

P{X 1 =1,X 2 =2,X 3 = 0 }=(.20)(.10)(.30)=.006

where we have letXidenote the change on theith day. ■