Discrete Mathematics for Computer Science

526 CHAPTER 8 Discrete Probability

Substituting px(x) = 2 p(w), where the summation is over all a) E? in the event (X =

x), gives

E(X)= L' (X IS"P6)

XEQX 0c-(X=x)

= S ( S: X.-P(W))

XEQX aOE(X=x)

= S S x~o) .p(w)

XCQx we(X=x)

Suppose that Qx = (xx.. x, }. Then, the double sum above can be written out as the

following sum of sums:

E X(o).p(w)±..+ + + ý X ).p((w).

(tE(X=xj) We(X=x,)

Since the events (X = xl). (X = x,) partition S2, each a) •? is contributing exactly

X ((o) p ((o) to this sum of sums. Hence,

E(X) = 5: X(CO) • p().

weQ

Example 5. Compute the expectation of the random variable X in Example 4 by using

Theorem 2.

Solution. Instead of computing the probability distribution px induced by X on Q^2 x, we

work directly with the original sample space 02.

E(X) = 5 p(0v)X(av)

= (-$l)p(l) + ($l)p(^2 ) + (-$^3 )p(^3 ) + ($1)p(4) + (-$5)p(^5 ) + ($l)p(6)

1 1 1 1 1 1

= (-$)- + (1)- + (-$3)- + ($1- + (-$5)- + ($)-

6 2 12 12 12 12

1

6

which agrees with our previous answer. U

8.75 The Sum of Random Variables

Suppose that X1 .. X, are random variables defined on some sample space S?. We can

use these to define new random variables on Q in various ways. For example, we can define

a random variable SUM by setting

SUM(cv) = XI (cO) + "" + Xn (a))

for each (0 E Q?.

Example 6. Suppose we toss three fair coins. We associate with this experiment a sample

space E2 of eight triples each having probability 1/8. The first, second, and third compo-

nents of each triple is 1 or 0 depending on whether the first, the second, or the third coin