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This fact is the basis of a hypothesis test, a "proportion z-test", for the value of p using x/n, the sample proportion and estimator of p, in a common test statistic. [35] For example, suppose one randomly samples n people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree ...
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [ 1 ] [ 2 ] In particular, the discrete-time Lyapunov equation (also known as Stein equation ) for X {\displaystyle X} is
The binomial test is useful to test hypotheses about the probability of success: : = where is a user-defined value between 0 and 1.. If in a sample of size there are successes, while we expect , the formula of the binomial distribution gives the probability of finding this value:
Lyapunov functions are used extensively in control theory to ensure different forms of system stability. The state of a system at a particular time is often described by a multi-dimensional vector. A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state.
Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium.
Within a system whose bins are filled according to the binomial distribution (such as Galton's "bean machine", shown here), given a sufficient number of trials (here the rows of pins, each of which causes a dropped "bean" to fall toward the left or right), a shape representing the probability distribution of k successes in n trials (see bottom of Fig. 7) matches approximately the Gaussian ...
Taking the biological population model as an example x n is a number between zero and one, which represents the ratio of existing population to the maximum possible population. [ May, Robert M. (1976) 2 ] This nonlinear difference equation is intended to capture two effects:
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).