Foundations of Data Science

Thus, n! ≤ n n e −n√ ne. For the lower bound on n!, start with the fact that for any
x 0 ≥ 1/2 and any real ρ
ln x 0 ≥ 1 2 (ln(x 0 + ρ) + ln(x 0 − ρ)) implies ln x 0 ≥
Thus,
ln(n!) = ln 2 + ln 3 + · · · + ln n ≥
∫ n+.5
x=1.5
from which one can derive a lower bound with a calculation.
Stirling approximation for the binomial coefficient
( n
( en
) k
≤
k)
k
∫ x0 +.5
x=x 0 −0.5
ln x dx,
ln x dx.
Using the Stirling approximation for k!,
( n n!
=
k)
(n − k)!k! ≤ nk
k! ∼ ( en
) k
= .
k
The gamma function
For a > 0
Γ (a) =
∫ ∞
0
x a−1 e −x dx
Γ ( 1
2)
=
√ π, Γ (1) = Γ (2) = 1, and for n ≥ 2, Γ (n) = (n − 1)Γ (n − 1) .
The last statement is proved by induction on n. It is easy to see that Γ(1) = 1. For n ≥ 2,
we use integration by parts.
∫
∫
f (x) g ′ (x) dx = f (x) g (x) − f ′ (x) g (x) dx
Write Γ(n) = ∫ ∞
x=0 f(x)g′ (x) dx, where, f(x) = x n−1 and g ′ (x) = e −x . Thus,
as claimed.
Γ(n) = [f(x)g(x)] ∞ x=0 +
∫ ∞
x=0
(n − 1)x n−2 e −x dx = (n − 1)Γ(n − 1),
Cauchy-Schwartz Inequality
( n∑
) ( n∑
) ( n∑
) 2
x 2 i yi
2 ≥ x i y i
i=1 i=1
i=1
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