Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 7] PROBABILITY 133

The set of ordered pairs(x 1 ,f(x 1 )),(x 2 ,f(x 2 )),...,(xt,f(xt))is called thedistributionof the random variable
X; it is usually given by a table as in Fig. 7-4. This functionfhas the following two properties:

(i)f(xk)≥0 and (ii)

∑

k

f(xk)= 1

ThusRXwith the above assignments of probabilities is a probability space. (Sometimes we will use the pair
notation[xk,pk]to denote the distribution ofXinstead of the functional notation[x, f (x)]).

Fig. 7-4 Distributionfof a random variableX

In the case thatSis an equiprobable space, we can easily obtain the distribution of a random variable from
the following result.

Theorem 7.8: LetSbe an equiprobable space, and letfbe the distribution of a random variableXonSwith
the range spaceRX={x 1 ,x 2 ,...,xt}. Then

pi=f(xi)=

number of points inSwhose image isxi

number of points inS

EXAMPLE 7.14 LetXbe the random variable in Example 7.13 which assigns the sum to the toss of a pair of
dice. Noten(S)=36, andRx={ 2 , 3 ,..., 12 }. Using Theorem 7.8, we obtain the distribution f ofXas follows:

f( 2 )= 1 / 36 ,since there is one outcome (1, 1) whose sum is 2.

f( 3 )= 2 / 36 ,since there are two outcomes, (1, 2) and (2,1), whose sum is 3.

f( 4 )= 3 / 36 ,since there are three outcomes, (1, 3), (2, 2) and (3, 1), whose sum is 4.

Similarly,f( 5 )= 4 /36,f( 6 )= 5 / 36 ,...,f( 12 )= 1 /36. Thus the distribution ofXfollows:

x 23456789101112

f(x) 1 /36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/ 36

Expectation of a Random Variable

LetXbe a random variable on a probability spaceS={s 1 ,s 2 ,...,sm}Then themeanorexpectationofX
is denoted and defined by:

μ=E(X)=X(s 1 )P (s 1 )+X(s 2 )P (sa 2 )+···+X(sm)P (sm)=

∑

X(sk)P (sk)

In particular, ifXis given by the distributionfin Fig. 7-4, then the expectation ofXis:

μ=E(X)=x 1 f(x 1 )+x 2 f(x 2 )+···+xtf(xt)=

∑

xkf(xk)

Alternately, when the notation[xk,pk]is used instead of[xk,f(xk)],

μ=E(X)=x 1 p 1 +x 2 p 2 +···+xtpt=

∑

xipi

(For notational convenience, we have omitted the limits in the summation symbol.)