The Poisson distribution is a discrete distribution that models the number of events based on a constant rate of occurrence. The Poisson distribution can be used as an approximation to the binomial when the number of independent trials is large and the probability of success is small.
The uniform distribution characterizes data over an interval uniformly, with a as the smallest value and b as the largest value. In This Topic Probability density function Binomial distribution Chi-square distribution Discrete distribution Exponential distribution F-distribution Geometric distribution. Integer distribution Lognormal distribution Normal distribution Poisson distribution t-distribution Uniform distribution Weibull distribution.
Probability density function The probability density function PDF of a random variable, X, allows you to calculate the probability of an event, as follows: For continuous distributions, the probability that X has values in an interval a, b is precisely the area under its PDF in the interval a, b. For discrete distributions, the probability that X has values in an interval a, b is exactly the sum of the PDF also called the probability mass function of the possible discrete values of X in a, b.
Use PDF to determine the value of the probability density function at a known value x of the random variable X. Binomial distribution The binomial distribution is used to represent the number of events that occurs within n independent trials. Notation Term Description n number of trials x number of events p event probability. Chi-square distribution If X has a standard normal distribution, X 2 has a chi-square distribution with one degree of freedom, allowing it to be a commonly used sampling distribution.
Formula The probability density function PDF is:. Discrete distribution A discrete distribution is one that you define yourself. Exponential distribution The exponential distribution can be used to model time between failures, such as when units have a constant, instantaneous rate of failure hazard function. F-distribution The F-distribution is also known as the variance-ratio distribution and has two types of degrees of freedom: numerator degrees of freedom and denominator degrees of freedom.
Geometric distribution. Finite Gaussians and finite quantum oscillators 2 2. Any d-dimensional Hilbert space is isomorphic to Cd , and can be regarded as the space of all the complex functions defined on a set with d elements. Here Zd is the ring of integers modulo d. Theorem 1. Direct consequence of the polynomial relation!
The Kravchuk functions Km satisfy the relation! It is known that the hypergeometric function! Theorem 3. Theorem 4. In this paper, we are mainly interested in the normalized Gaussians see Fig. The function G3 takes positive as well as negative values, but the corresponding distribution of probability G3 2 has the shape of a Gaussian function see Fig.
The Wigner functions corresponding to g1 , g2 and g3 are sums of products of finite Gaussians see Fig. Finite Gaussians and finite quantum oscillators 15 Proof.
Finite Gaussians and finite quantum oscillators 17 5. It is Fourier invariant, and one can prove that the eigenspaces corresponding to its eigenvalues are one-dimensional. The normalized eigenfunctions h0 , h1 , Finite Gaussians and finite quantum oscillators 19 0. Finite Gaussians and finite quantum oscillators 21 Table 1. On the existence of revivals When the dimension d becomes larger and larger, the eigenvalues of the considered finite oscillators have the tendency to become equidistant see Fig.
On the other hand, the existence of some equidistant energy levels implies the existence of revivals. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. However, as any positive integrable function, it is proportional to another PDF, which happens to be itself gaussian. The rest is calculus.
There is no probabilistic interpretation to this algebraic fact that I am aware of and, to tell you the truth, I wonder why this factoid was selected as noticeable on the website you link to much more significant are the characteristic function of a gaussian PDF being a gaussian function and the convolution of gaussian PDFs being a gaussian PDF.
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