Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 801 — #809
19.2. Independence 801
Likewise ŒM D 1 is the event fT T T; HHH g and has probability 1=4.
More generally, any assertion about the values of random variables defines an
event. For example, the assertion that C 1 defines
ŒC 1 D fT T T; T TH; TH T; H T T g;
and so PrŒC 1 D 1=2.
Another example is the assertion that C M is 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,
ŒC M is odd D fHHH g:
19.2 Independence
The notion of independence carries over from events to random variables as well.
Random variables R 1 and R 2 are independent iff for all x 1 ; x 2 , the two events
ŒR 1 D x 1 and ŒR 2 D x 2
are independent.
For example, are C and M independent? Intuitively, the answer should be “no.”
The number of heads C completely determines whether all three coins match; that
is, whether M D 1. But, to verify this intuition, we must find some x 1 ; x 2 2 R
such that:
PrŒC D x 1 AND M D x 2 ¤ PrŒC D x 1 PrŒM D x 2 :
One appropriate choice of values is x 1 D 2 and x 2 D 1. In this case, we have:
PrŒC D 2 AND M D 1 D 0 ¤ 1 4 3 8
D PrŒM D 1 PrŒC D 2:
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, let H 1 be the indicator variable for the event that the first flip
is a Head, so
ŒH 1 D 1 D fHHH; H TH; HH T; H T T g: