Unformatted text preview: P (A) = P (B ) = P (C ) = 1/2. The
intersection of any two of these events is
A \ B = B \ C = C \ A = {HHH, T T T }
an event of probability 1/4. From this it follows that
P (A \ B ) = 11
1
= · = P (A)P (B )
4
22 i.e., A and B are independent. Similarly, B and C are independent and C and
A are independent; so A, B , and C are pairwise independent. The three events
A, B , and C are not independent, however, since A \ B \ C = {HHH, T T T }
and hence
✓ ◆3
1
1
P (A \ B \ C ) = 6=
= P (A)P (B )P (C )
4
2
The last example is somewhat unusual. However, the moral of the story is that
to show several events are independent, you have to check more than just that
each pair is independent. A.2. RANDOM VARIABLES, DISTRIBUTIONS A.2 211 Random Variables, Distributions Formally, a random variable is a realvalued function deﬁned on the sample
space. However, in most cases the sample space is usually not visible, so we
describe the random variables by giving their distributions. In the discrete
case where the random variable can take on...
View
Full
Document
 Spring '10
 DURRETT
 The Land

Click to edit the document details