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.

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive