Probability Generating Function Research Articles

A stochastic approach to the electron avalanche problem

A probability balance equation is formulated, based upon the regeneration point method, for the number of electrons at timet, positionz in an avalanche initiated by a trigger electron. The result is given in the form of a nonlinear integral equation for the probability-generating function. From this equation we can obtain equations for the mean value, which has been studied by many authors, and also the mean square value and higher moments. In the special case of 1/v absorption, scattering and ionization cross-sections, a complete expression for the probability is obtained in the infinite-medium, time-dependent case. The spatial development of the avalanche is studied and, by using some results obtained recent by Boffiet al., we can obtain a concise expression for Townsend’s coefficient.

On linearly regressive processes

We consider an important class of integer valued stochastic processes characterised by a certain form of time dependent probability generating function (P.G.F.). Certain results are obtained concerning the correlation structure of such processes, and it is shown that these are considerably simplified in the case of a stationary distribution having been attained. The results are illustrated in the latter case using some commonly occurring processes belonging to the class under consideration.

On linearly regressive processes

We consider an important class of integer valued stochastic processes characterised by a certain form of time dependent probability generating function (P.G.F.). Certain results are obtained concerning the correlation structure of such processes, and it is shown that these are considerably simplified in the case of a stationary distribution having been attained. The results are illustrated in the latter case using some commonly occurring processes belonging to the class under consideration.

Photoelectron statistics due to mixing of different types of fields

The probability-generating functions for the number of electrons ejected from a photodetector by an incident light beam, which is a mixture of coherent signals and other kinds of light beams, are arrived at by using a generalized cumulant expansion method. It is found that a knowledge of the statistical correlation properties of the mixing beams is necessary to deduce the resulting distributions.

Dead-time correction for any distribution of input events

A recently developed relation is used to calculate the counting-loss in the case when one element (at the front-end or after a prescaler) of the instrumentation used is of the paralysable type. The only necessary information about the input train of events is its probability-generating function.

A theorem on discrete time branching processes allowing immigration

We consider a population which evolves at discrete points in time by branching and immigration, and in which each member reproduces independently of all others. Let Fn(x) denote the probability generating function (P.G.F.) of the number of offspring produced by each member of the nth generation, Bn–1(x) the P.G.F. of the number of immigrants joining the nth generation and Zn the population size in the nth generation. We write

A theorem on discrete time branching processes allowing immigration

We consider a population which evolves at discrete points in time by branching and immigration, and in which each member reproduces independently of all others. Let Fn (x) denote the probability generating function (P.G.F.) of the number of offspring produced by each member of the nth generation, Bn– 1(x) the P.G.F. of the number of immigrants joining the nth generation and Zn the population size in the nth generation. We write

Population growth and the multi-type Galton-Watson process.

THE general Galton–Watson (GW) process with n types, Ti(i=1, 2 … n), is specified1 by the type of the individual starting the process at time (generation) zero, and the vector-valued probability generating function (PGF) where the scalar quantity fi(s) is the multi-type PGF of the numbers of individuals of the various types produced by an individual of type Ti (i = 1 … n).

Erratum: Power spectrum of a pulse train specified by its probability-generating function

Erratum: Power spectrum of a pulse train specified by its probability-generating function

Power spectrum of a pulse train specified by its probability-generating function

A stationary stochastic train of nonoverlapping pulses is considered, and a link is established between the conditional probability density pk(τ) for the appearance of k successive count-to-count intervals during the time interval τ, assuming that the first count-to-count interval has begun (with the first pulse) at t = 0, and that the kth interval has ended (with the last pulse) at t = τ, and the probability Gj(τ) that j successive pulses of this train will appear during the same time interval τ (supposing that the initial time t = 0 has been selected at random). If the train considered consists of identical unit pulses, it is also possible to relate to Gj(τ) its autocorrelation function R(τ), as well as its power density P(ω).

Some properties of the 'Hermite' distribution

SUMMARY The Hermite distribution is the generalized Poisson distribution whose probability generating function (P.G.F.) is exp [a1(s - 1) + a2(s2 - 1)]. The probabilities (and factorial moments) can be conveniently expressed in terms of modified Hermite polynomials (hence the proposed name). Writing these in confluent hypergeometric form leads to two quite different representations of any particular probability-one a finite series, the other an infinite series. The cumulants and moments are given. Necessary conditions on the parameters and their maximum-likelihood estimation are discussed. It is shown, with examples, that the Hermite distribution is a special case of the Poisson Binomial distribution (n = 2) and may be regarded as either the distribution of the sum of two correlated Poisson variables or the distribution of the sum of an ordinary Poisson variable and an (independent) Poisson 'doublet' variable (i.e. the occurrence of pairs of events is distributed as a Poisson). Lastly its use as a penultimate limiting form of distributions with P.G.F.

Transient Phenomena in branching stochastic processes

The following scheme for generating particles is considered. Each particle existing at a given time splits up into k particles with the probability $\delta _{k1} + p_k \Delta t + o(\Delta t),\delta _{11} = 1,\delta _{k1} = 0$ for $k \ne 1$, in time $\Delta t \to 0$ independent of its age and origin, as well as the history of the other particles. We define the probability-generating function by $f(x) = \Sigma _{k = 0}^\infty p_k x^k $ and denote factorial moments by $a = f'(1)$, $b = f''(1)$, $c = f'''(1)$. Let $K(b_0 ,c_0 )$ be a class of $f(x)$ with $f''(1) \geqq b_0 > 0$ and $f'''(1) \leqq c_0 < \infty $. Let $\mu _t $ be the number of particles at time t. We introduce the function \[ q(t;a) = \left\{ \begin{gathered} \frac{{e^{at} }} {{1 + \frac{b} {{2a}}\left( {e^{at} - 1} \right)}}\quad {\text{for }}a \ne 0, \hfill \frac{1} {{1 + \frac{b} {2}t}}\quad {\text{for }}a = 0 \hfill \end{gathered} \right. .\]The following asymptotic formula for $t \to \infty ,a \to 0$ holds true uniformly for all $f(x...

Branching Stochastic Processes for Particles Diffusing in a Bounded Domain with Absorbing Boundaries

Particles diffusing in a multidimensional restricted domain with absorbing boundaries produce independent new particles according to a scheme for branching processes with the probability-generating function $F(z) = \sum _{k = 0}^\infty P_{k^{z^k } } $, where $P_k $ is the conditional probability that a particle turns into k particles, if at all. Let $P_n (x,t)$ be the probability that a particle existing at point x after t generations turns into n particles. The probability-generating function \[ F(x,t,z)\mathop \sum \limits_n P_n (x,t)z^n \] satisfies (11). Let $P_0 (x) = \lim _{t \to 0} P_0 (x,t) $ be the probability of extinction if the initial particle was at point x. $P_0 (x)$ satisfies a non-linear integral equation of Hammerstein’s type (21). There is a critical value $A_0 > 1$ for $A = F'$ (1). If $A \leqq A_0 $, then $P_0 (x) \equiv 1$; if $A > A_0 $, then $P_0 (x)< 1 $. The critical value is \[ A_0 = \frac{{D_{\lambda 1} + c}}{c}, \] where D is the coefficient of diffusion, ${1 / c}$ is the mean...

Alternative expressions for probability-generating functions concerning an inherited character after a panmixia

Alternative expressions for probability-generating functions concerning an inherited character after a panmixia