Skellam Probability mass function
Examples of the probability mass function for the Skellam distribution. The horizontal axis is the index n. (Note that the function is only defined at integer values of n. The connecting lines do not indicate continuity.) | Cumulative distribution function
| | Parameters | | | Support | | | pmf | | | cdf | | | Mean | μ1 − μ2 | | Median | N/A | | Mode | | | Variance | μ1 + μ2 | | Skewness | | | Kurtosis | 1 / (μ1 + μ2) | | Entropy | | | mgf | | | Char. func. | | The Skellam distribution is the discrete probability distribution of the difference N1 − N2 of two correlated or uncorrelated random variables N1 and N2 having Poisson distributions with different expected values μ1 and μ2. It is useful in describing the statistics of the difference of two images with simple photon noise, as well as describing the point spread distribution in certain sports where all scored points are equal, such as baseball, hockey and soccer. Download high resolution version (1300x975, 225 KB) Some probability mass functions for the Skellam distribution File links The following pages link to this file: Probability distribution Skellam distribution Categories: User-created public domain images ...
In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...
In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or...
In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds...
In probability theory and statistics, the median is a number that separates the highest half of a sample, a population, or a probability distribution from the lowest half. ...
In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ...
In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ...
In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ...
Entropy of a Bernoulli trial as a function of success probability. ...
In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith...
Some mathematicians use the phrase characteristic function synonymously with indicator function. The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A...
In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ...
A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...
In probability theory and statistics, the Poisson distribution is a discrete probability distribution (discovered by Siméon-Denis Poisson (1781–1840) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilité des jugements en matières criminelles et matière civile [Research on the...
In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds...
In physics, the number of photons collected by an instrument which are emitted from any incoherent source are distributed according to a Poisson distribution, as long as the average intensity is constant over the bandwidth of the instrument. ...
Spread betting is a form of gambling on the outcome of any event where the more accurate the gamble, the more is won and conversely the less accurate the more is lost. ...
Baseball is a team sport, in which a fist-sized ball is thrown by a defensive player called a pitcher and hit by an offensive player called a batter with a round, smooth stick called a bat. ...
Ice hockey, known simply as hockey in areas where it is more common than field hockey, is a team sport played on ice. ...
The striker (wearing red jersey) has run past the defender (in white jersey) and is about to take a shot at the goal, while the goalkeeper positions himself to stop the ball. ...
Only the case of uncorrelated variables will be considered in this article. See Karlis & Ntzoufras, 2003 for the use of the Skellam distribution to describe the difference of correlated Poisson-distributed variables.
Recall that probability mass function of a Poisson distribution with mean μ is given by In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...
The Skellam probability mass function is the cross-covariance of two Poisson distributions: (Skellam, 1946) In signal processing, the cross-covariance (or sometimes cross correlation) is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. ...
where I k(z) is the modified Bessel function of the first kind. The above formulas have assumed that any term with a negative factorial is set to zero. The special case for μ1 = μ2 is given by (Irwin, 1937): In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number α (the order). ...
Note also that, using the limiting values of the Bessel function for small arguments, we can recover the Poisson distribution as a special case of the Skellam distribution for μ2=0.
Properties The Skellam probability mass function is of course normalized: We know that the generating function for a Poisson distribution is: In probability theory, the probability-generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. ...
In probability theory and statistics, the Poisson distribution is a discrete probability distribution (discovered by Siméon-Denis Poisson (1781–1840) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilité des jugements en matières criminelles et matière civile [Research on the...
It follows that the generating function G(t |μ1,μ2) for a Skellam probability function will be: Notice that the form of the generating function implies that the distribution of the sums or the differences or, in fact, any linear combination of two Skellam-distributed variables are again Skellam-distributed. The moment-generating function is given by: In probability theory, the probability-generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. ...
In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith...
which yields the raw moments mk . Define: Then the raw moments mk are The central moments M k are In probability theory and statistics, the kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X − E[X])k], where E is the expectation operator. ...
The mean, variance, skewness, and kurtosis excess are respectively: In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds...
In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ...
In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ...
The cumulant-generating function is given by: Cumulants of probability distributions In probability theory and statistics, the cumulants κn of a probability distribution are given by where X is any random variable whose probability distribution is the one whose cumulants are taken. ...
which yields the cumulants: Cumulants of probability distributions In probability theory and statistics, the cumulants κn of a probability distribution are given by where X is any random variable whose probability distribution is the one whose cumulants are taken. ...
For the special case when μ1 = μ2, an asymptotic expansion of the modified Bessel function of the first kind yields for large μ: In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...
In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number α (the order). ...
(Abramowitz & Stegun 1972, p. 377). Also, for this special case, when n is also large, and of order of the square root of 2μ, the distribution tends to a normal distribution: The Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ...
The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. ...
These special results can easily be extended to the more general case of different means.
References - Abramowitz, M. and Stegun, I. A. (Eds.). 1972. Modified Bessel functions I and K. Sections 9.6–9.7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, pp. 374–378. New York: Dover.
- Irwin, J. O. 1937. The frequency distribution of the difference between two independent variates following the same Poisson distribution. Journal of the Royal Statistical Society: Series A 100 (3): 415–416.
- Karlis, D. and Ntzoufras, I. 2003. Analysis of sports data using bivariate Poisson models. Journal of the Royal Statistical Society: Series D (The Statistician) 52 (3): 381–393. doi:10.1111/1467-9884.00366
- Karlis, D. and Ntzoufras, J. Bayesian analysis of paired count data. Unpublished manuscript. http://www.ba.aegean.gr/ntzoufras/papers/11_poisdif.pdf
- Skellam, J. G. 1946. The frequency distribution of the difference between two Poisson variates belonging to different populations. Journal of the Royal Statistical Society: Series A 109 (3): 296.
|
Feeds |
Contact