In the Democratic Republic of the Congo, girls are outnumbered two to one.
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. 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.Graph each system of inequalities.
Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region. A "system" of linear inequalities is a set of linear inequalities that you deal with all at once.
Usually you start off with two or three linear inequalities. The technique for solving these systems is fairly simple.
kcc1 Count to by ones and by tens. kcc2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1). kcc3 Write numbers from 0 to Represent a number of objects with a written numeral (with 0 representing a count of no objects).
kcc4a When counting objects, say the number names in the standard order, pairing each . Graph functions, plot data, evaluate equations, explore transformations, and much more – for free! Search using a saved search preference or by selecting one or more content areas and grade levels to view standards, related Eligible .
Learn how to graph systems of two-variable linear inequalities, like "y>x-8 and y.