Elementary Probability Theory and UsefulMathematical Tools(1)
Seminario de Fsica Estadstica - MHCF
C. A. Rivera et al.
Museo Histrico de Ciencias FsicasE.A.P de Fsica
Facultad de Ciencias Fsicas
Universidad Nacional Mayor de San Marcos
8 de Mayo / 2015
C. A. Rivera et al. (MHCF-UNMSM) Seminario de Fsica Estadstica 8 de Mayo / 2015 1 / 23
Overview
1 Introduction
2 Permutations and Combinations
3 Definition of Probability
4 Fourier Series and Fourier Transforms
5 The Dirac "Function"
6 Stochastic Variables and Probability
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Introduction
Thermodynamics not explain the behavior of microscopic systemswith a large number of degrees of freedom.Probability is a concept that must be introduce.Because the probability distribution are defined for variables, thatcan be discrete or continuous stochastic variables.
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In this fashion stochastic means that the system can not bedefined intrinsically of a deterministic way.A interesting result of thermodynamics is the existence of highlycorrelated stochastics quantities: many systems have the samecritical exponents, regardless of the microscopic structure of thesystem.
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Permutations and Combinations
1 Addition principle: operations are mutually exclusive. m + nways.
2 Multiplication principle: operations can be done one after theother. mxn ways.
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Permutation: is a arrangement of a set of N distint objects in adefinite order.
N(N 1)(N 2)x ...x1 = N!
PNR = N(N 1)x ...x(N R + 1) =N!
(N R)!N!
n1!n2!...nk !
Where n1 + n2 + ...+ nk = N
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Combination: is a arrangement of a set of N distint objectswithout regard to order.
R!CNR = PNR
CNR =N!
(N R)!R!
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C. A. Rivera et al. (MHCF-UNMSM) Seminario de Fsica Estadstica 8 de Mayo / 2015 8 / 23
Definition of Probability
Probability is a quantization of our expectation of the outcome of anevent or experiment. P(A), the probability for a event A . In N identicalevents: NP(A). If (N ) we expect that NP(A) will result in theoutcome A.
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A sample space of an experiment is a set, S, of elements such thatany outcome of the experiments corresponds to one or more elementsof the set.An event is a subset of a sample space S of an experiment.
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The probability of an event A can be found by using the followingprocedure:
Set up a sample space S of all possible outcomes.Assign probabilities to the elements of the sample space.To obtain P(A) add the probabilities assigned to elements of thesubset of S that corresponds to A.
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In working with probabilities, some ideas from set theory are useful.For example, P(A B) denote the probability that both events A and B.Also,
P(A B) = P(A) + P(B) P(A B)
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If the two events are exclusive,
P(A B) = P(A) + P(B)
If events A1,A2, ...,Am are mutually exclusive and exhaustive, thenA1 A2 ... Am = S and the m events form a partition of the samplespace S into m subsets. If A1,A2, ...,Am form a partition, then
P(A1) + P(A2) + ...+ P(Am) = 1
The events A and B are independent if and only if
P(A B) = P(A)P(B)Independent events are not mutually exclusive events.
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The conditional probability P(B|A) gives us the probability that event Aoccurs as the result of an experiment if B also occurs.
P(B|A) = P(A B)P(B)Also:
P(A)P(A|B) = P(B)P(B|A)Since, P(A B) = P(B A).If A and B are independent, then
P(B|A) = P(A)
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Fourier Series
Periodic functions. f (x + L) = f (x)For the exponential:
e2ipixL = cos(2pixL ) isin(2pi
x
L )
Let f (x) be a periodic function with a fundamental periof of L , if itsatisfies certain mathematical conditions (as is practically always thecase in physics) it can be expanded in a series of imaginaryexponentials or trigonometric functions.
f (x) =+
n=
cneiknx
with kn = n 2piL
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The coefficients cn of the Fourier series are given by the formula,
cn =1L
x0+Lx0
dxeiknx f (x)
The set of values |cn| is called the Fourier spectrum of f (x).Also, f (x) can be expressed as,
f (x) = c0 ++n=1
(cneiknx + cne
iknx )
f (x) = a0 ++n=1
(ancos(knx) + bnsin(knx))
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Fourier Transforms
The Fourier integrals (transforms) are interpreted as a limit of a FourierSeries, consider a function f (x) which is not necessarily periodic.
F (k) = 12pi
+
dxeikx f (x)
f (x) = 12pi
+
dkeikxF (K )
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The Dirac "Function"The Dirac "function" is actually a distribution. However, like mostphysicists, we shall treat it like an ordinary function.
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For any function f (x) defined at the origin,
+
dx(x)f (x) = f (0)
More generally, (x x0), is defined by:
+
dx(x x0)f (x) = f (x0)
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Stochastic Variables and Probability
In order to apply probability theory to the real world, we must introducethe concept of stochastic, or random, variable. A quantity whose valueis a number determined by the outcome of an experiment is called astochastic variable X .A stochastic variable X , on the space S, is a function which mapselements of S into the set of real numbers {R} and viceversa.
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Discrete Stochastic Variable
X has a countable set of realizations {xi}. If S is a probability space,so we can assign a probability pi to each realization xi .A probability function PX (x) is defined as,
PX (x) =n
i=1pi(x xi)
And a distribution density function FX (x), defined as,
FX (x) = x
dyPX (y) =n
i=1pi(x xi)
Where is a Heaviside function.
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Continuous Stochastic Variables
Let X be a stochastic variable which can take on a continuous set ofvalues, such as an interval on the real axis. An interval of {a x b}corresponds to an event.
FX (x) = x
dyPX (y)
It is the probability to find the stochastic variable X in the interval{ x}Also if Y = H(x), the probability density for the stochatic variable Y ,PY (y)] , is defined as
PY (y) =
+
dx(y H(x))PX (x)
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IntroductionPermutations and CombinationsDefinition of ProbabilityFourier Series and Fourier TransformsThe Dirac "Function"Stochastic Variables and Probability
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