Transition Matrices of Markov Chains

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

Suppose that if it is sunny today, there is a 60% chance it will be sunny tomorrow and that if it is not sunny today, there is a 20% chance it will be sunny tomorrow. Use the four transition probabilities sunnysunny, sunnynot sunny, not sunnysunny, and not sunnynot sunny to form the transition matrix .

[more]

If we assume today's sunniness depends only on yesterday's sunniness (and not on previous days), then this system is an example of a Markov Chain, an important type of stochastic process. Powers of the transition matrix can be used to compute the long-term probability of the system being in either of the two states. As the power grows, the entries in the first row will all approach the long term probability that the system is in the first state (sunny).

If it is sunny today, there is about a 1/3 chance of sun in five days. If it is cloudy today, there is about a 1/3 chance of sun in five days. Thus, today's weather doesn't matter for the long-term prediction!

[less]

Contributed by: Ed Pegg Jr (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send