Foundations of Data Science

from which the current inequality follows.
Arithmetic and geometric means
The arithmetic mean of a set of nonnegative reals is at least their geometric mean.
For a 1 , a 2 , . . . , a n > 0,
1
n∑
a i ≥
n
n√ a 1 a 2 · · · a n .
i=1
Assume that a 1 ≥ a 2 ≥ . . . ≥ a n . We reduce the proof to the case when all the a i
are equal using the variational method; in this case the inequality holds with equality.
Suppose a 1 > a 2 . Let ε be a positive infinitesimal. Add ε to a 2 and subtract ε from a 1 to
get closer to the case when they are equal. The left hand side 1 n
∑ n
i=1 a i does not change.
(a 1 − ε)(a 2 + ε)a 3 a 4 · · · a n = a 1 a 2 · · · a n + ε(a 1 − a 2 )a 3 a 4 · · · a n + O(ε 2 )
> a 1 a 2 · · · a n
for small enough ε > 0. Thus, the change has increased n√ a 1 a 2 · · · a n . So if the inequality
holds after the change, it must hold before. By continuing this process, one can make all
the a i equal.
Approximating sums by integrals
For monotonic decreasing f(x),
∫n+1
x=m
f (x)dx ≤
n∑
f (i) ≤
i=m
∫ n
x=m−1
f (x)dx.
See Fig. 12.1. Thus,
∫n+1
x=2
1
dx ≤
x 2
n∑
1
= 1 + 1 + · · · + 1 ≤
i 2 4 9 n 2
i=2
and hence 3 − 1 ≤ ∑ n 1
≤ 2 − 1 .
2 n+1 i 2 n
i=1
n∫
x=1
1
dx
x 2
Jensen’s Inequality
For a convex function f,
( )
1
f
2 (x 1 + x 2 ) ≤ 1 2 (f (x 1) + f (x 2 )) .
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