18.2. Independence 741
LikewiseŒMD1çis the eventfT T T;HHHgand has probability1=4.
More generally, any assertion about the values of random variables defines an
event. For example, the assertion thatC 1 defines
ŒC1çDfT T T;T TH;THT;HT Tg;
and so PrŒC1çD1=2.
Another example is the assertion thatCMis an odd number. If you think about
it for a minute, you’ll realize that this is an obscure way of saying that all three
coins came up heads, namely,
ŒCMis oddçDfHHHg:
18.2 Independence
The notion of independence carries over from events to random variables as well.
Random variablesR 1 andR 2 areindependentiff for allx 1 ;x 2 , the two events
ŒR 1 Dx 1 ç and ŒR 2 Dx 2 ç
are independent.
For example, areCandMindependent? Intuitively, the answer should be “no.”
The number of heads,C, completely determines whether all three coins match; that
is, whetherM D 1. But, to verify this intuition, we must find somex 1 ;x 22 R
such that:
PrŒCDx 1 ANDMDx 2 ç¤PrŒCDx 1 çPrŒMDx 2 ç:
One appropriate choice of values isx 1 D 2 andx 2 D 1. In this case, we have:
PrŒCD 2 ANDMD1çD 0 ¤
1
4
3
8
DPrŒMD1çPrŒCD2ç:
The first probability is zero because we never have exactly two heads (C D 2 )
when all three coins match (M D 1 ). The other two probabilities were computed
earlier.
On the other hand, letH 1 be the indicator variable for the event that the first flip
is a Head, so
ŒH 1 D1çDfHHH;HTH;HHT;HT Tg:
