# Connexions

You are here: Home » Content » Applied Probability » Appendix B to Applied Probability: some mathematical aids

## Navigation

### Table of Contents

• Preface to Pfeiffer Applied Probability

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Rice Digital Scholarship

This collection is included in aLens by: Digital Scholarship at Rice University

Click the "Rice Digital Scholarship" link to see all content affiliated with them.

#### Also in these lenses

• UniqU content

This collection is included inLens: UniqU's lens
By: UniqU, LLC

Click the "UniqU content" link to see all content selected in this lens.

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection:

Collection by: Paul E Pfeiffer. E-mail the author

# Appendix B to Applied Probability: some mathematical aids

Module by: Paul E Pfeiffer. E-mail the author

Summary: A variety of mathematical aids to probability analysis and calculations.

## Series

• 1. : Geometric series From the expression (1-r)(1+r+r2+...+rn)=1-rn+1(1-r)(1+r+r2+...+rn)=1-rn+1, we obtain
k=0nrk=1-rn+11-rforr1k=0nrk=1-rn+11-rforr1
(1)
For |r|<1|r|<1, these sums converge to the geometric series k=0rk=11-rk=0rk=11-r
Differentiation yields the following two useful series:
k=1krk-1=1(1-r)2for|r|<1andk=2k(k-1)rk-2=2(1-r)3for|r|<1k=1krk-1=1(1-r)2for|r|<1andk=2k(k-1)rk-2=2(1-r)3for|r|<1
(2)
For the finite sum, differentiation and algebraic manipulation yields
k=0nkrk-1=1-rn[1+n(1-r)](1-r)2whichconvergesto1(1-r)2for|r|<1k=0nkrk-1=1-rn[1+n(1-r)](1-r)2whichconvergesto1(1-r)2for|r|<1
(3)
• 2. : Exponential series. ex=k=0xkk!ande-x=k=0(-1)kxkk!foranyx ex=k=0xkk!ande-x=k=0(-1)kxkk!foranyx
Simple algebraic manipulation yields the following equalities useful for the Poisson distribution:
k=nkxkk!=xk=n-1xkk!andk=nk(k-1)xkk!=x2k=n-2xkk!k=nkxkk!=xk=n-1xkk!andk=nk(k-1)xkk!=x2k=n-2xkk!
(4)
• 3. : Sums of powers of integers i = 1 n i = n ( n + 1 ) 2 i = 1 n i 2 = n ( n + 1 ) ( 2 n + 1 ) 6 i = 1 n i = n ( n + 1 ) 2 i = 1 n i 2 = n ( n + 1 ) ( 2 n + 1 ) 6

## Some useful integrals

• 1. : The gamma functionΓ(r)=0tr-1e-tdtforr>0Γ(r)=0tr-1e-tdtforr>0
Integration by parts shows Γ(r)=(r-1)Γ(r-1)forr>1Γ(r)=(r-1)Γ(r-1)forr>1
By induction Γ(r)=(r-1)(r-2)(r-k)Γ(r-k)forr>kΓ(r)=(r-1)(r-2)(r-k)Γ(r-k)forr>k
For a positive integer n,Γ(n)=(n-1)!withΓ(1)=0!=1n,Γ(n)=(n-1)!withΓ(1)=0!=1
• 2. : By a change of variable in the gamma integral, we obtain
0tre-λtdt=Γ(r+1)λr+1r>-1,λ>00tre-λtdt=Γ(r+1)λr+1r>-1,λ>0
(5)
• 3. : A well known indefinite integral gives
ate-λtdt=1λ2e-λa(1+λa)andat2e-λatdt=1λ3e-λa[1+λa+(λa)2/2]ate-λtdt=1λ2e-λa(1+λa)andat2e-λatdt=1λ3e-λa[1+λa+(λa)2/2]
(6)
For any positive integer m,
atme-λtdt=m!λm+1e-λa1+λa+(λa)22!++(λa)mm!atme-λtdt=m!λm+1e-λa1+λa+(λa)22!++(λa)mm!
(7)
• 4. : The following integrals are important for the Beta distribution.
01ur(1-u)sdu=Γ(r+1)Γ(s+1)Γ(r+s+2)r>-1,s>-101ur(1-u)sdu=Γ(r+1)Γ(s+1)Γ(r+s+2)r>-1,s>-1
(8)
For nonnegative integers m,n01um(1-u)ndu=m!n!(m+n+1)!m,n01um(1-u)ndu=m!n!(m+n+1)!

## Some basic counting problems

We consider three basic counting problems, which are used repeatedly as components of more complex problems. The first two, arrangements and occupancy are equivalent. The third is a basic matching problem.

1. Arrangements of r objects selected from among n distinguishable objects.
1. The order is significant.
2. The order is irrelevant.
For each of these, we consider two additional alternative conditions.
1. No element may be selected more than once.
2. Repitition is allowed.
2. Occupancy of n distinct cells by r objects. These objects are
1. Distinguishable.
2. Indistinguishable.
The occupancy may be
1. Exclusive.
2. Nonexclusive (i.e., more than one object per cell)

The results in the four cases may be summarized as follows:

1. Ordered arrangements, without repetition (permutations). Distinguishable objects, exclusive occupancy.
P(n,r)=n!(n-r)!P(n,r)=n!(n-r)!
(9)
2. Ordered arrangements, with repitition allowed. Distinguishable objects, nonexclusive occupancy.
U(n,r)=nrU(n,r)=nr
(10)
1. Arrangements without repetition, order irrelevant (combinations). Indistinguishable objects, exclusive occupancy.
C(n,r)=n!r!(n-r)!=P(n,r)r!C(n,r)=n!r!(n-r)!=P(n,r)r!
(11)
2. Unordered arrangements, with repetition. Indistinguishable objects, nonexclusive occupancy.
S(n,r)=C(n+r-1,r)S(n,r)=C(n+r-1,r)
(12)
3. Matchingn distinguishable elements to a fixed order. Let M(n,k)M(n,k) be the number of permutations which give k matches.

### Example 1: n = 5 n = 5

Natural order 1 2 3 4 5

Permutation 3 2 5 4 1 (Two matches– positions 2, 4)

We reduce the problem to determining m(n,0)m(n,0), as follows:

1. Select k places for matches in C(n,k)C(n,k) ways.
2. Order the n-kn-k remaining elements so that no matches in the other n-kn-k places.
M(n,k)=C(n,k)M(n-k,0)M(n,k)=C(n,k)M(n-k,0)
(13)
Some algebraic trickery shows that M(n,0)M(n,0) is the integer nearest n!/en!/e. These are easily calculated by the MATLAB command M = round(gamma(n+1)/exp(1)) For example
>> M = round(gamma([3:10]+1)/exp(1));
>> disp([3:6;M(1:4);7:10;M(5:8)]')
3           2           7        1854
4           9           8       14833
5          44           9      133496
6         265          10     1334961


## Extended binomial coefficients and the binomial series

• The ordinary binomial coefficient is C(n,k)=n!k!(n-k)!C(n,k)=n!k!(n-k)! for integers n>0,0knn>0,0kn
For any real x, any integer k, we extend the definition by
C(x,0)=1,C(x,k)=0fork<0,andC(n,k)=0forapositiveintegerk>nC(x,0)=1,C(x,k)=0fork<0,andC(n,k)=0forapositiveintegerk>n
(14)
and
C(x,k)=x(x-1)(x-2)(x-k+1)k!otherwiseC(x,k)=x(x-1)(x-2)(x-k+1)k!otherwise
(15)
Then Pascal's relation holds: C(x,k)=C(x-1,k-1)+C(x-1,k)C(x,k)=C(x-1,k-1)+C(x-1,k)
The power series expansion about t=0t=0 shows
(1+t)x=1+C(x,1)t+C(x,2)t2+x,-1<t<1(1+t)x=1+C(x,1)t+C(x,2)t2+x,-1<t<1
(16)
For x=nx=n, a positive integer, the series becomes a polynomial of degree n.

## Cauchy's equation

1. Let f be a real-valued function defined on (0,)(0,), such that
1. f(t+u)=f(t)+f(u)fort,u>0f(t+u)=f(t)+f(u)fort,u>0, and
2. There is an open interval I on which f is bounded above (or is bounded below).
Then f(t)=f(1)tt>0f(t)=f(1)tt>0
2. Let f be a real-valued function defined on (0,)(0,) such that
1. f(t+u)=f(t)f(u)t,u>0f(t+u)=f(t)f(u)t,u>0, and
2. There is an interval on which f is bounded above.
Then, either f(t)=0fort>0f(t)=0fort>0 , or there is a constant a such that f(t)=eatfort>0f(t)=eatfort>0

[For a proof, see Billingsley, Probability and Measure, second edition, appendix A20]

## Countable and uncountable sets

A set (or class) is countable iff either it is finite or its members can be put into a one-to-one correspondence with the natural numbers.

### Examples

• The set of odd integers is countable.
• The finite set {n:1n1000}{n:1n1000} is countable.
• The set of all rational numbers is countable. (This is established by an argument known as diagonalization).
• The set of pairs of elements from two countable sets is countable.
• The union of a countable class of countable sets is countable.

A set is uncountable iff it is neither finite nor can be put into a one-to-one correspondence with the natural numbers.

### Examples

• The class of positive real numbers is uncountable. A well known operation shows that the assumption of countability leads to a contradiction.
• The set of real numbers in any finite interval is uncountable, since these can be put into a one-to-one correspondence of the class of all positive reals.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

### Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

### Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

### Add:

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks