A The Basic Formalism of Probability Theory in Visual Studio .NET

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Appendix A The Basic Formalism of Probability Theory
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of independent trials necessary to obtain m occurrences of an event which has constant probability p of occurring at each trial Then we have P ( = m + k) = for k = 1, 2, 3,
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A1325 Poisson Distribution The family is indexed by parameter > 0 Then random variable distribution when the probability distribution is given by
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m+k 1 m 1
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p m (1 p)k
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(A30)
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has the Poisson
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P ( = k) =
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e k , k!
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(A31)
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for k = 1, 2, 3, The Poisson distribution is the limit of a sequence of binomial distributions in which n tends to in nity and p tends to zero such that n p (binomial mean) remains equal to (Poisson mean) [221]
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A1326 Poisson Process [222] A stochastic process {N (t), t v 0} is said to be a counting process if N (t) represents the total number of events that have occurred up to time t and satis es the following conditions:
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(i) N (t) v 0 (ii) N (t) is integer-valued (iii) If s < t, then N (s) u N (t) (iv) For s < t, N (t) N (s) equals the number of events that have occurred in the interval (s, t) A counting process possesses independent increments if the number of events which occur in disjoint time intervals are independently distributed A counting process is a Poisson process with rate , > 0, if it satis es the following conditions: (i) N (0) = 0 (ii) The process has independent increments (iii) The number of events in any interval of length t has the Poisson distribution with mean t
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A13 Some Important Distributions
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A1327 Mixtures of Discrete Probability Distributions [223] Consider a family of discrete distributions indexed by a parameter For random variable , write P ( = x) = p x ( ) Let G(d ) represent a probability measure on Then the probability distribution
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P r( = x) =
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p x ( ) G(d )
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is described as p x ( ) mixed on by G(d ) For example, if p x ( ) is the Poisson distribution and G(d ) is the exponential distribution, then we have the result that has the geometrical distribution More generally, a Poisson variate mixed on by the gamma distribution (see below) has the negative binomial distribution
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A1328 Beta Distribution The beta distribution is a continuous probability distribution with the probability density function de ned on the interval [0, 1]:
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1 x a 1 (1 x)b 1 , B(a, b)
(A32)
where a and b are parameters that must be greater than zero, and B(a, b) = (a) (b)/ (a + b) is the beta function expressed in terms of the gamma functions
A1329 Wigner Semicircle Distribution The Wigner semicircle distribution is a continuous probability distribution with the interval [ R, R] as support The graph of its probability density function f is a semicircle of radius R centered at the origin (actually a semiellipse with suitable normalization):
f (x) =
2 1 x2 , R2
(A33)
for R < x < R, and f (x) = 0 if x > R or x < R This distribution arises as the limiting distribution of eigenvalues of many random symmetric matrices as the size of the matrix approaches in nity
A13210 Gamma Distribution The gamma distribution is a continuous probability distribution Its probability density function can be expressed in terms of the gamma function:
f (x; k, ) =
e x/ x k 1 , k (k)
(A34)
where k > 0 is the shape parameter and > 0 is the scale parameter
Appendix A The Basic Formalism of Probability Theory
A13211 Normal Distribution The normal distribution is certainly the most popular among continuous probability distributions It is also called the Gaussian distribution It is actually a family of distributions of the same general form, differing only in their location and scale parameters: the mean and standard deviation Its probability density function with mean and standard deviation (equivalently, variance 2 ) is given by
f (x; , ) =
1 2 2 e (x ) /2 2
(A35)
The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one Because the graph of its probability density resembles a bell, it is often called a bell curve Approximately normal distributions occur in many situations, as a result of the central limit theorem When there is reason to suspect the presence of a large number of small effects acting additively and independently, it is reasonable to assume that observations will be normal There are statistical methods to empirically test that assumption Two examples of the occurrence of the normal law in physics:
Photon counts Light intensity from a single source varies with time, and is usually assumed to be normally distributed However, quantum mechanics interprets measurements of light intensity as photon counting Ordinary light sources that produce light by thermal emission should follow a Poisson distribution or Bose Einstein distribution on very short time scales On longer time scales (longer than the coherence time), the addition of independent variables yields an approximately normal distribution The intensity of laser light, which is a quantum phenomenon, has an exactly normal distribution Measurement errors Repeated measurements of the same quantity are expected to yield results which are clustered around a particular value If all major sources of errors have been taken into account, it is assumed that the remaining error must be the result of a large number of very small additive effects, and hence normal Deviations from normality are interpreted as indications of systematic errors which have not been taken into account Note that this is the central assumption of the mathematical theory of errors A13212 Cauchy Distribution The Cauchy distribution is an example of a continuous distribution that does not have an expected value or a variance In physics it is usually called a Lorentzian, and it is the distribution of the energy of an unstable state in quantum mechanics In particle physics, the extremely short lived particles associated with such unstable states are called resonances The Cauchy distribution has the probability density