Discrete Mathematics for Computer Science

Discrete Random Variables 527

lands heads or tails, respectively. For a triple w E 2, let X 1 (w) be 1 if the first coin lands

heads up and 0 if otherwise. Define X 2 (a) and X 3 ((0) in similar fashion for the second and

the third coins, respectively. If the first coin lands heads and the other two land tails, what

is the value of SUM on that outcome?

Solution. Since o = (1, 0, 0), we have X, ((o) = l and X 2 (6o) = X 3 (o9) = 0. Hence,

SUM(wO) = 1+0+0= 1 0

Example 7. What is the expectation of SUM in the preceding example?

Solution. The range Q^2 x of X is the number of heads that can show, so QŽx = 10, 1, 2, 3}.

The probability distribution induced by X on Q^2 x is

1

Px(O) = p({(0, 0,

0))) = -

8

3

Px(1) = p({(1, 0, 0), (0, 1, 0), (0, 0, 1)}) = 8

px(^2 ) = p({(1, 1, 0), (1, 0, 1), (0, 1, 1)}) = 8

1

px(^3 ) = p({(1, 1, 1)1) = -

8

Hence,

E(SUM) = x. Px(x)

1 3 3 1

8 8 8 8

3

2

We now establish notation for some common random variables that are defined in

terms of one or more others. The symbols for the new random variables will indicate how

they are computed from the old ones rather than simply being single letters like X. When

we are given several random variables X1. X, we will assume they are all defined on

the same sample space QŽ.

The sum of random variables. The random variable that sends ca to

Xt(c) + .. + Xn(CO)

is denoted by

(Xi +.. + X")

Its expectation is denoted by E(X 1 + •.. -+- Xv). Hence, the random variable SUM in the
preceding example would be denoted (X 1 + X 2 + X 3 ). The value of

(XI +- X2 +• +' ' Xn)

at a particular co is denoted by

(X1 + X2 .+ Xn)(cO)