180 exponent

)

56

–

54

1

–

bn

1

2–m = –

–

2m

For example, in tossing a pair of dice, the expected

sum is:

(The possible sums are the numbers 2 through 12, with

the probability of casting 2 being 1/36, a 3 being 2/36,

and so on.)

Expected value is usually denoted by the letter μfrom

the G

REEK ALPHABET

. If an experiment has an infinite

number of possible outcomes, then the expected value

is given by an infinite sum (

SERIES

). The

BINOMIAL DIS

-

TRIBUTION

is an example of this. If the random phe-

nomenon can produce a continuous array of values (for

example, the height of an individual can be any value,

including fractional ones), then the expected value is

given by an

INTEGRAL

:

μ= ∫∞

–∞x p(x)dx

Here p(x) is the probability density function of the

random variable under consideration for the given

DISTRIBUTION

.

The notion of expected value was first developed

by Dutch scientist Christiaan Huygens (1629–95) in his

treatise On Reasoning in a Dice Game.

See also

HISTORY OF PROBABILITY AND STATISTICS

(essay).

exponent (index) For a real number band a posi-

tive whole number m, the shorthand bmis used for

the repeated multiplication of bwith itself mtimes:

bm= b×b×…×b(mtimes). Thus, for example, 25=

2 ×2 ×2 ×2 ×2 = 32, (–1)3= (–1) ×(–1) ×(–1) = –1,

, and 101= 10. In the expression bm,

mis called the exponent, or the index, and bis called

the base of the exponent. We also call bma power

of b.

The product of two expressions bmand bnwith the

same base bis itself a repeated multiplication of the

number b. Precisely:

This establishes the multiplication rule for exponents:

To multiply two expressions with the same

base, retain the common base and add together

the exponents: bm×bn= bm+ n.

The power rule for exponents, (bm)n= bmn, follows.

(One must add mwith itself ntimes.) The multiplica-

tion rule is considered fundamental and allows us to

define exponential quantities bmfor values of mother

than whole numbers. We follow the principle that the

multiplication rule is to always hold.

Consider, for example, the expression 20. This

quantity has no meaning when interpreted as “the mul-

tiplication of two with itself zero times.” However, one

can assign a meaningful value to this expression by

multiplying it with another power of two. For example,

according to the multiplication rule, it must be the case

that 20×25= 20+5 = 25. This says that 20×32 = 32,

which tells us that 20must equal one. The multiplica-

tion rule thus leads to the rule:

The zero exponent for any nonzero base equals

1: b0= 1.

To make sense of the quantity 2–1, again invoke the

multiplication rule. We have, for example, 2–1 ×23=

2–1+3 = 22. This reads: 2–1 ×8 = 4. It must be the case

then that 2–1 = 1/2. Similar calculations show that 2–2

must equal 1/4, and that 2–3 must equal 1/8. In general,

. This works for any nonzero base b.

A negative exponent indicates that a reciprocal

must be taken: b–n= .

We can make use of this observation to compute ,

for example. Rewriting, we have

.

To divide two expressions with the same

base, retain the common base and subtract

the exponents: = bn–m

bm

–

bn