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218 7. Approximations and Inequalities

Comment: It turns out that

when the function f(t) is continuous in t.

Let us get some insight into this result.

Since

c ( L >

tk(l - tyk = 1

(why?), we can think of the terms of this sum as nonnegative weights

which add up to 1. For each t, B(f, n; t) is a weighted average of the

values f(O), f(lln), fCJ/n),... , f( 1). When t = 0, the only weight

which does not vanish is at the k = 0 term and B(f, n;O) = f(0).

When t = 1, all the weight is at the k = n term and B(f, n; 1) = f(1).

For intermediate values of t, it turns out that the terms where k/n

is close to t are weighted more heavily than the other terms. By the

continuity of j(t), when k/n is near t, then f(k/n) is close to f(t),

so that, in the average, the most weighted terms are those that are

about equal to f(t). Th is weighting becomes more pronounced as n

increases, so that B(f, n;t) draws ever closer to f(t) for each value

oft.

Another way to look at the situation is as follows. Fix t between 0

and 1. Imagine a dartboard of area 1, of which a portion of area t

is painted red. Hurl n darts randomly at the board, all piercing the

board somewhere. If k of them land in the red area, you receive a

prize of f(k/n) d o 11 am. What is your expectation (intuitively, your

average winnings) if this game were to be frequently repeated? The

probability of getting exactly k darts in the red area is the product

of

n

( k >

the number of ways of selecting the k darts from the n thrown

tk the probability that all k of these darts land in the red area

(1 - t)n-k the probability that the other (n - k) darts do not land

in the red area.

The expectation is the average payoff over all possible occurrences,

i.e. the sum of the products of the payoffs and probability of getting

them. This is B(f, n; t).

Now increase the number n of darts used in the game. For very large

n, it is highly probable that the proportion landing in the red area

will be close to t, so that the expected payoff will be close to f(t).