Foundations of Data Science

Miscellaneous integrals
∫ 1
x=0
x α−1 (1 − x) β−1 dx = Γ(α)Γ(β)
Γ(α + β)
For definition of the gamma function see Section 12.3 Binomial coefficients
The binomial coefficient ( )
n
k =
n!
is the number of ways of choosing k items from n.
(n−k)!k!
The number of ways of choosing d + 1 items from n + 1 items equals the number of ways
of choosing the d + 1 items from the first n items plus the number of ways of choosing d
of the items from the first n items with the other item being the last of the n + 1 items.
( ( ) ( )
n n n + 1
+ = .
d)
d + 1 d + 1
The observation that the number of ways of choosing k items from 2n equals the
number of ways of choosing i items from the first n and choosing k − i items from the
second n summed over all i, 0 ≤ i ≤ k yields the identity
k∑
( )( ) ( )
n n 2n
= .
i k − i k
i=0
Setting k = n in the above formula and observing that ( ) (
n
i = n
n−i)
yields
More generally k ∑
i=0
12.3 Useful Inequalities
1 + x ≤ e x for all real x.
n∑
( ) 2 n
=
i
i=0
( 2n
n
)
.
( n m
) (
i)(
k−i = n+m
)
k by a similar derivation.
One often establishes an inequality such as 1 + x ≤ e x by showing that the difference
of the two sides, namely e x − (1 + x), is always positive. This can be done
by taking derivatives. The first and second derivatives are e x − 1 and e x . Since e x
is always positive, e x − 1 is monotonic and e x − (1 + x) is convex. Since e x − 1 is
monotonic, it can be zero only once and is zero at x = 0. Thus, e x − (1 + x) takes
on its minimum at x = 0 where it is zero establishing the inequality.
(1 − x) n ≥ 1 − nx for 0 ≤ x ≤ 1
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