We will assign a number to a line, which we call slope, that will give us a measure of the "steepness" or "direction" of the line. It is often convenient to use a special notation to distinguish between the rectan- gular coordinates of two different points.

The system's state space and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v.

Countable state space Continuous or general state space Discrete-time discrete-time Markov chain on a countable or finite state space Harris chain Markov chain on a general state space Continuous-time Any continuous stochastic process with the Markov property, e.

Usually the term "Markov chain" is reserved for a process with a discrete set of times, i. Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs.

Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term. While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: Besides time-index and state-space parameters, there are many other variations, extensions and generalizations see Variations.

For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise. The changes of state of the system are called transitions. The process is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state or initial distribution across the state space.

By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not terminate. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps.

Formally, the steps are the integers or natural numbersand the random process is a mapping of these to states. Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future.

From any position there are two possible transitions, to the next or previous integer. The transition probabilities depend only on the current position, not on the manner in which the position was reached.

For example, the transition probabilities from 5 to 4 and 5 to 6 are both 0. These probabilities are independent of whether the system was previously in 4 or 6. Another example is the dietary habits of a creature who eats only grapes, cheese, or lettuce, and whose dietary habits conform to the following rules: It eats exactly once a day.

If it ate cheese today, tomorrow it will eat lettuce or grapes with equal probability. It will not eat lettuce again tomorrow. This creature's eating habits can be modeled with a Markov chain since its choice tomorrow depends solely on what it ate today, not what it ate yesterday or any other time in the past.

One statistical property that could be calculated is the expected percentage, over a long period, of the days on which the creature will eat grapes.

A series of independent events for example, a series of coin flips satisfies the formal definition of a Markov chain.

However, the theory is usually applied only when the probability distribution of the next step depends non-trivially on the current state.

History[ edit ] Andrey Markov studied Markov chains in the early 20th century.

Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest inand a branching process, introduced by Francis Galton and Henry William Watson inpreceding the work of Markov.Graph inequalities In order to graph an inequality we work in 3 steps: First we graph our boundaries; we dash the line if the values on the line are not included in the boundary.

Improve your math knowledge with free questions in "Write inequalities from graphs" and thousands of other math skills. Answer to The graph of g(x) is the result of translating the graph of f(x) = {1/2}x three units to the left.

What is the equation of g(x)? A beautiful, free online graphing calculator from benjaminpohle.com View and Download Texas Instruments TI PLUS - Graphing Calculator manual book online.

Guidebook. TI PLUS - Graphing Calculator Calculator pdf manual download. Also for: Ti . candidate's examination number the united republic of tanzania national examinations council form two secondary education examination.

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