Foundations of Data Science

n+1 ∫
x=m
f (x)dx ≤
n ∑
i=m
f (i) ≤
n∫
x=m−1
f (x)dx
m − 1 m n n + 1
Figure 12.1: Approximating sums by integrals
More generally for any convex function f,
( n∑
)
f α i x i ≤
i=1
n∑
α i f (x i ),
∑
where 0 ≤ α i ≤ 1 and n α i = 1. From this, it follows that for any convex function f and
random variable x,
i=1
i=1
E (f (x)) ≥ f (E (x)) .
We prove this for a discrete random variable x taking on values a 1 , a 2 , . . . with Prob(x =
a i ) = α i :
E(f(x)) = ∑ i
α i f(a i ) ≥ f
( ∑
i
α i a i
)
= f(E(x)).
f(x 1 )
f(x 2 )
x 1 x 2
Figure 12.2: For a convex function f, f ( )
x 1 +x 2
2 ≤
1
(f (x 2 1) + f (x 2 )) .
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