Estadística Intro

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


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.


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.


Permutations and Combinations

1 Addition principle: operations are mutually exclusive. m + nways.

2 Multiplication principle: operations can be done one after theother. mxn ways.


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


Combination: is a arrangement of a set of N distint objectswithout regard to order.

R!CNR = PNR

CNR =N!

(N R)!R!


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.


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.


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.


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)


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.


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


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))


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 )


The Dirac "Function"The Dirac "function" is actually a distribution. However, like mostphysicists, we shall treat it like an ordinary function.


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)


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.


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.


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)


IntroductionPermutations and CombinationsDefinition of ProbabilityFourier Series and Fourier TransformsThe Dirac "Function"Stochastic Variables and Probability

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