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A variety of mathematical aids to probability analysis and calculations.

Series

  • Geometric series From the expression ( 1 - r ) ( 1 + r + r 2 + . . . + r n ) = 1 - r n + 1 , we obtain
    k = 0 n r k = 1 - r n + 1 1 - r for r 1
    For | r | < 1 , these sums converge to the geometric series k = 0 r k = 1 1 - r
    Differentiation yields the following two useful series:
    k = 1 k r k - 1 = 1 ( 1 - r ) 2 for | r | < 1 and k = 2 k ( k - 1 ) r k - 2 = 2 ( 1 - r ) 3 for | r | < 1
    For the finite sum, differentiation and algebraic manipulation yields
    k = 0 n k r k - 1 = 1 - r n [ 1 + n ( 1 - r ) ] ( 1 - r ) 2 which converges to 1 ( 1 - r ) 2 for | r | < 1
  • Exponential series . e x = k = 0 x k k ! and e - x = k = 0 ( - 1 ) k x k k ! for any x
    Simple algebraic manipulation yields the following equalities usefulfor the Poisson distribution:
    k = n k x k k ! = x k = n - 1 x k k ! and k = n k ( k - 1 ) x k k ! = x 2 k = n - 2 x k k !
  • 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

Some useful integrals

  • The gamma function Γ ( r ) = 0 t r - 1 e - t d t for r > 0
    Integration by parts shows Γ ( r ) = ( r - 1 ) Γ ( r - 1 ) for r > 1
    By induction Γ ( r ) = ( r - 1 ) ( r - 2 ) ( r - k ) Γ ( r - k ) for r > k
    For a positive integer n , Γ ( n ) = ( n - 1 ) ! with Γ ( 1 ) = 0 ! = 1
  • By a change of variable in the gamma integral, we obtain
    0 t r e - λ t d t = Γ ( r + 1 ) λ r + 1 r > - 1 , λ > 0
  • A well known indefinite integral gives
    a t e - λ t d t = 1 λ 2 e - λ a ( 1 + λ a ) and a t 2 e - λ a t d t = 1 λ 3 e - λ a [ 1 + λ a + ( λ a ) 2 / 2 ]
    For any positive integer m ,
    a t m e - λ t d t = m ! λ m + 1 e - λ a 1 + λ a + ( λ a ) 2 2 ! + + ( λ a ) m m !
  • The following integrals are important for the Beta distribution.
    0 1 u r ( 1 - u ) s d u = Γ ( r + 1 ) Γ ( s + 1 ) Γ ( r + s + 2 ) r > - 1 , s > - 1
    For nonnegative integers m , n 0 1 u m ( 1 - u ) n d u = 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 ) !
      2. Ordered arrangements, with repitition allowed. Distinguishable objects, nonexclusive occupancy.
        U ( n , r ) = n r
      1. Arrangements without repetition, order irrelevant ( combinations ). Indistinguishable objects, exclusive occupancy.
        C ( n , r ) = n ! r ! ( n - r ) ! = P ( n , r ) r !
      2. Unordered arrangements, with repetition. Indistinguishable objects, nonexclusive occupancy.
        S ( n , r ) = C ( n + r - 1 , r )
  3. Matching n distinguishable elements to a fixed order. Let M ( n , k ) 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)

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

  1. Select k places for matches in C ( n , k ) ways.
  2. Order the n - k remaining elements so that no matches in the other n - k places.
    M ( n , k ) = C ( n , k ) M ( n - k , 0 )
    Some algebraic trickery shows that M ( n , 0 ) 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 ( n , k ) = n ! k ! ( n - k ) ! for integers n > 0 , 0 k n
    For any real x , any integer k , we extend the definition by
    C ( x , 0 ) = 1 , C ( x , k ) = 0 for k < 0 , and C ( n , k ) = 0 for a positive integer k > n
    and
    C ( x , k ) = x ( x - 1 ) ( x - 2 ) ( x - k + 1 ) k ! otherwise
    Then Pascal's relation holds: C ( x , k ) = C ( x - 1 , k - 1 ) + C ( x - 1 , k )
    The power series expansion about t = 0 shows
    ( 1 + t ) x = 1 + C ( x , 1 ) t + C ( x , 2 ) t 2 + x , - 1 < t < 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 ( 0 , ) , such that
    1. f ( t + u ) = f ( t ) + f ( u ) for t , 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 ) t t > 0
  2. Let f be a real-valued function defined on ( 0 , ) such that
    1. f ( 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 ) = 0 for t > 0 , or there is a constant a such that f ( t ) = e a t for 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 { n : 1 n 1000 } 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.

Questions & Answers

what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
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Kyle
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Adin
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
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sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
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characteristics of micro business
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for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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
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s. Reply
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.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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