Memoryless property

Memoryless Property A geometric r.v. has a property of forgetting the past in the following sense Given that the first success has not occurred by the sth trial, the probability that it will not occur for at least another t trials is the same as if one were to start all over that is, it is the probability that the first success will not occur for at least t trials starting at time zero. This properly is easy to prove ... 

Memoryless Property The exponential distribution shares the memoryless property with the geometric distribution, and it is the only continuous distribution having this property. The proof of the property is simple ... 

One might think that the second probabiUty should be smaller, since the person arrived after the previous bus had left and so there is a smaller chance that he will have to wait for more than 10 minutes. However, because of memoryless property of the exponential distribution... 

It follows from this that the CV of the exponential distribution is 1. The exponential distribution has the remarkable memoryless property, which states that... 

When the chain does leave state i, it chooses its next state j i according to the probabilities exponentially distributed Because the Markov property of X implies that T must have the memoryless property (13), and this in turn implies that T has the exponential distribution. 

From the memoryless properties of the feasible sets, transition probabilities, and rewards, it is intuitive that it should be sufficient to consider memoryless deterministic policies. This can be shown to be true for finite horizon problems of the form (34). 

It appears reasonable that a device that is neither wearing out nor improving with age has the memoryless property. This can be shown mathematically. If the failure rate is constant, then... 

As discussed in a previous section, constant rate processes are random events with the memoryless property. This probability distribution appears weU suited to modeling the failures of high-quality devices operating in a benign environment. The constant rate probability distribution, however, does not appear to be well suited to such events as system reconfiguration. Some of these procedures, for example, are fixed time programs. If the system is halfway through a 10-ms procedure, then it is known that the procedure... 

In state S A, the memoryless property of a constant rate process is used to construct the transition to state SC. When it arrives in state SA, device B does not remember having spent some time in state S. Hence, for device B, the transition out of state SA is the same as the transition out of state S. If device B were a component that wore out, then the transition from state S A to SC in Fig. 21.19 would depend on the amount of time the system spent in state S. The modeling and computation would be more difficult. A similar discussion holds for state S B. In state S C, both devices have failed. 

The distribution of the waiting time until first occurrence, given that we have already waited until time t is the same as the distribution of the waiting time starting from 0. The exponential distribution is the only continuous distribution that has this memoryless property. 

In this chapter we introduce Markov chains. These are a special type of stochastic process, which are processes that move around a set of possible values where the future values can t be predicted with certainty. There is some chance element in the evolution of the process through time. The set of possible values is called the state space of the process. Markov chains have the "memoryless" property that, given the past and present states, the future state only depends on the present state. This chapter will give us the necessary background knowledge about Markov chains that we will need to understand Markov chain Monte Carlo sampling. 

A series of probable transitions between states can be described with Markov modeling. The natural course of a disease, for example, can be viewed for an individual subject as a sequence of certain states of health (12). A Markovian stochastic process is memoryless. To predict what the future state will be, knowledge of the current state is sufficient and is independent of where the process has been in the past. This is termed the strong Markov property (13). 

Understanding the structure and function of biomolecules requires insight into both thermodynamic and kinetic properties. Unfortunately, many of the dynamical processes of interest occur too slowly for standard molecular dynamics (MD) simulations to gather meaningful statistics. This problem is not confined to biomolecular systems, and the development of methods to treat such rare events is currently an active field of research. - If the kinetic system can be represented in terms of linear rate equations between a set of M states, then the complete spectrum of M relaxation timescales can be obtained in principle by solving a memoryless master equation. This approach was used in the last century for a number of studies involving atomic... 

As we saw in Section 2, Poisson arrivals may be thought of as completely random arrivals in a very exact sense. They are neither too regular, as for example if they came once every five minutes, nor too irregular, as for example if they came in batches of between 1 and 20 with highly variable interarrival times. The Poisson process implies memorylessness, in that observations of past arrivals give no hint as to future arrivals. To expand on this property, if the arrivals to the real system have been unusually heavy recently, this is no indication that they will continue to be heavy, nor is it an indication that they will become less heavy. If actual arrivals tend to come in clusters, or if the arrival rate fluctuates throughout the workday, the Poisson model may be inappropriate. The qualitative... 

As an example of deriving the density function from the properties of the event, this section will show that memoryless events must have an exponential density function. The derivation uses the result from real analysis that the only continuous function h(x) with the property, h a + b) = h a)h b) is the exponential function h t) = for some a. 

The next example considers the failure of three identical components with failure rate a. This example uses the basic properties of competing events, sequential events, and memoryless processes. Figure 21.23 presents the model for the failure of at least one component. State A2 represents the failure of one, two, or three components. In Fig. 21.23(b) state B2 represents the failure of exactly one component, whereas state B3 represent the failure of two or three components. Figure 21.23(c) gives all of the details. Choosing the middle diagram in Fig. 21.23(b), the differential equations are... 

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