Take a guided tour through the powerful mathematics and statistics used to model the chaos of the financial markets.
A Markov chain is a mathematical system that evolves with time. It consists of a set of states and a matrix that contains the probabilities of transitioning from one state to another. In each time unit, the chain transitions randomly from one state to another, making the state the chain is in at any time at random variable.
For example, a person walking along the integers who moves to the left or right with equal probability is a Markov chain. The state space is the integers, and the transition probabilities are \(0.5\) for adjacent states, and 0 for all others.
In the Markov chain pictured, what is the probability that a process beginning at state A will be at state B after \(2\) moves?
Let \(k\) represent the number of days until a person turns \(200\) years old. What is the biggest problem with this model of life?
Any general stochastic process can be made to satisfy the Markov property by altering the state space (and adding probabilities for any new states). In this way, it can "turn into" a Markov chain. Given a stochastic process with state space \(S\), which of the following methods would create a Markov chain that models the same process?
The Golden State Warriors are strongly affected by morale, and they win games according to the probabilities in the following table:
|States||Win Tomorrow||Lose Tomorrow|
If they lose the first two games in a "best of \(7\)" (first to \(4\)) series, what is the probability that they will be able to come back and win it?