# 17.1 Appendix b to applied probability: some mathematical aids

 Page 1 / 1
A variety of mathematical aids to probability analysis and calculations.

## Series

• Geometric series From the expression $\left(1-r\right)\left(1+r+{r}^{2}+...+{r}^{n}\right)=1-{r}^{n+1}$ , we obtain
$\sum _{k=0}^{n}{r}^{k}=\frac{1-{r}^{n+1}}{1-r}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}r\ne 1$
For $|r|<1$ , these sums converge to the geometric series $\sum _{k=0}^{\infty }{r}^{k}=\frac{1}{1-r}$
Differentiation yields the following two useful series:
$\sum _{k=1}^{\infty }k{r}^{k-1}=\frac{1}{{\left(1-r\right)}^{2}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}|r|<1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\sum _{k=2}^{\infty }k\left(k-1\right){r}^{k-2}=\frac{2}{{\left(1-r\right)}^{3}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}|r|<1$
For the finite sum, differentiation and algebraic manipulation yields
$\sum _{k=0}^{n}k{r}^{k-1}=\frac{1-{r}^{n}\left[1+n\left(1-r\right)\right]}{{\left(1-r\right)}^{2}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{which}\phantom{\rule{4.pt}{0ex}}\text{converges}\phantom{\rule{4.pt}{0ex}}\text{to}\phantom{\rule{0.277778em}{0ex}}\frac{1}{{\left(1-r\right)}^{2}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}|r|<1$
• Exponential series . ${e}^{x}=\sum _{k=0}^{\infty }\frac{{x}^{k}}{k!}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{e}^{-x}=\sum _{k=0}^{\infty }{\left(-1\right)}^{k}\frac{{x}^{k}}{k!}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{any}\phantom{\rule{0.277778em}{0ex}}x$
Simple algebraic manipulation yields the following equalities usefulfor the Poisson distribution:
$\sum _{k=n}^{\infty }k\frac{{x}^{k}}{k!}=x\sum _{k=n-1}^{\infty }\frac{{x}^{k}}{k!}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\sum _{k=n}^{\infty }k\left(k-1\right)\frac{{x}^{k}}{k!}={x}^{2}\sum _{k=n-2}^{\infty }\frac{{x}^{k}}{k!}$
• Sums of powers of integers $\sum _{i=1}^{n}i=\frac{n\left(n+1\right)}{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\sum _{i=1}^{n}{i}^{2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$

## Some useful integrals

• The gamma function $\Gamma \left(r\right)={\int }_{0}^{\infty }{t}^{r-1}{e}^{-t}\phantom{\rule{0.166667em}{0ex}}dt\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}r>0$
Integration by parts shows $\Gamma \left(r\right)=\left(r-1\right)\Gamma \left(r-1\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}r>1$
By induction $\Gamma \left(r\right)=\left(r-1\right)\left(r-2\right)\cdots \left(r-k\right)\Gamma \left(r-k\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}r>k$
For a positive integer $n,\phantom{\rule{0.277778em}{0ex}}\Gamma \left(n\right)=\left(n-1\right)!\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{with}\phantom{\rule{0.277778em}{0ex}}\Gamma \left(1\right)=0!=1$
• By a change of variable in the gamma integral, we obtain
${\int }_{0}^{\infty }{t}^{r}{e}^{-\lambda t}\phantom{\rule{0.166667em}{0ex}}dt=\frac{\Gamma \left(r+1\right)}{{\lambda }^{r+1}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}r>-1,\phantom{\rule{0.277778em}{0ex}}\lambda >0$
• A well known indefinite integral gives
${\int }_{a}^{\infty }t{e}^{-\lambda t}\phantom{\rule{0.166667em}{0ex}}dt=\frac{1}{{\lambda }^{2}}\phantom{\rule{0.166667em}{0ex}}{e}^{-\lambda a}\phantom{\rule{0.166667em}{0ex}}\left(1+\lambda a\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\int }_{a}^{\infty }{t}^{2}{e}^{-\lambda at}\phantom{\rule{0.166667em}{0ex}}dt=\frac{1}{{\lambda }^{3}}\phantom{\rule{0.166667em}{0ex}}{e}^{-\lambda a}\phantom{\rule{0.166667em}{0ex}}\left[1+\lambda a+{\left(\lambda a\right)}^{2}/2\right]$
For any positive integer m ,
${\int }_{a}^{\infty }{t}^{m}{e}^{-\lambda t}\phantom{\rule{0.166667em}{0ex}}dt=\frac{m!}{{\lambda }^{m+1}}\phantom{\rule{0.166667em}{0ex}}{e}^{-\lambda a}\left[1,+,\lambda ,a,+,\frac{{\left(\lambda a\right)}^{2}}{2!},+,\cdots ,+,\frac{{\left(\lambda a\right)}^{m}}{m!}\right]$
• The following integrals are important for the Beta distribution.
${\int }_{0}^{1}{u}^{r}{\left(1-u\right)}^{s}\phantom{\rule{0.166667em}{0ex}}du=\frac{\Gamma \left(r+1\right)\Gamma \left(s+1\right)}{\Gamma \left(r+s+2\right)}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}r>-1,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}s>-1$
For nonnegative integers $m,n\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\int }_{0}^{1}{u}^{m}{\left(1-u\right)}^{n}\phantom{\rule{0.166667em}{0ex}}du=\frac{m!n!}{\left(m+n+1\right)!}$

## 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.
$P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$
2. Ordered arrangements, with repitition allowed. Distinguishable objects, nonexclusive occupancy.
$U\left(n,r\right)={n}^{r}$
1. Arrangements without repetition, order irrelevant ( combinations ). Indistinguishable objects, exclusive occupancy.
$C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}=\frac{P\left(n,r\right)}{r!}$
2. Unordered arrangements, with repetition. Indistinguishable objects, nonexclusive occupancy.
$S\left(n,r\right)=C\left(n+r-1,r\right)$
3. Matching n distinguishable elements to a fixed order. Let $M\left(n,k\right)$ be the number of permutations which give k matches.

## $n=5$

Natural order 1 2 3 4 5

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

We reduce the problem to determining $m\left(n,0\right)$ , as follows:

1. Select k places for matches in $C\left(n,k\right)$ ways.
2. Order the $n-k$ remaining elements so that no matches in the other $n-k$ places.
$M\left(n,k\right)=C\left(n,k\right)M\left(n-k,0\right)$
Some algebraic trickery shows that $M\left(n,0\right)$ is the integer nearest $n!/e$ . 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 148335 44 9 133496 6 265 10 1334961

## Extended binomial coefficients and the binomial series

• The ordinary binomial coefficient is $C\left(n,k\right)=\frac{n!}{k!\left(n-k\right)!}$ for integers $n>0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0\le k\le n$
For any real x , any integer k , we extend the definition by
$C\left(x,0\right)=1,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}C\left(x,k\right)=0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}k<0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}C\left(n,k\right)=0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{a}\phantom{\rule{4.pt}{0ex}}\text{positive}\phantom{\rule{4.pt}{0ex}}\text{integer}\phantom{\rule{0.277778em}{0ex}}k>n$
and
$C\left(x,k\right)=\frac{x\left(x-1\right)\left(x-2\right)\cdots \left(x-k+1\right)}{k!}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{otherwise}$
Then Pascal's relation holds: $C\left(x,k\right)=C\left(x-1,k-1\right)+C\left(x-1,k\right)$
The power series expansion about $t=0$ shows
${\left(1+t\right)}^{x}=1+C\left(x,1\right)t+C\left(x,2\right){t}^{2}+\cdots \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}x,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}-1
For $x=n$ , a positive integer, the series becomes a polynomial of degree n .

## Cauchy's equation

1. Let f be a real-valued function defined on $\left(0,\infty \right)$ , such that
1. $f\left(t+u\right)=f\left(t\right)+f\left(u\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}t,\phantom{\rule{0.277778em}{0ex}}u>0$ , and
2. There is an open interval I on which f is bounded above (or is bounded below).
Then $f\left(t\right)=f\left(1\right)t\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}t>0$
2. Let f be a real-valued function defined on $\left(0,\infty \right)$ such that
1. $f\left(t+u\right)=f\left(t\right)f\left(u\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}t,\phantom{\rule{0.277778em}{0ex}}u>0$ , and
2. There is an interval on which f is bounded above.
Then, either $f\left(t\right)=0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}t>0$ , or there is a constant a such that $f\left(t\right)={e}^{at}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}t>0$

[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 $\left\{n:1\le n\le 1000\right\}$ 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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive By OpenStax By Joanna Smithback By Anonymous User By Wey Hey By David Corey By Lakeima Roberts By Jams Kalo By OpenStax By Rhodes By OpenStax