Table of Contents

## What is the first passage time problem?

The first-passage time problem of a general stochastic birth-death process involves finding the distribution of times for which the random process first passes through a specific threshold.

## What is first passage time distribution?

The first passage time distribution, h(t) ≡ −dS(t)/dt, is such that h(t)dt is the probability that a particle passes through the point b for the first time in the time interval (t, t + dt). It is usually difficult to solve the combined initial value and boundary value problem described by Eqs.

**What is mean recurrence time?**

The mean recurrence times of (countable state) irreducible and positive recurrent Markov chains are the spanning tree invariants of the first return loop systems. Henceforth, for (countable state) irreducible and positive recurrent Markov chains, spanning tree invariants of loop systems are mean recurrence times.

**What is the transition probability?**

the probability of moving from one state of a system into another state. If a Markov chain is in state i, the transition probability, pij, is the probability of going into state j at the next time step.

### Is chess fully observable?

Examples: Chess – the board is fully observable, and so are the opponent’s moves.

### Is Tic Tac Toe stochastic?

Tic-tac-toe: fully observable, deterministic, very small. Chess: fully observable, deterministic, very big. Monopoly: fully observable, stochastic, very big. Card games: stochastic, not fully observable, typically big.

**What is the formula of transition probability?**

The formulas for the transition probabilities are p11(t) = eat, p12(t) = bueft + qeat, and so on. In general, if i is a death state (that is, an absorbing state) then pii(t) = 1.

**What is the formula for transition probability?**

Recall that the elements of the transition matrix P are defined as: (P)ij = pij = P(X1 = j |X0 = i) = P(Xn+1 = j |Xn = i) for any n. pij is the probability of making a transition FROM state i TO state j in a SINGLE step.

#### Is chess deterministic or strategic?

Environment | Observable | Deterministic |
---|---|---|

Chess with a clock | Fully | Strategic |

Chess without a clock | Fully | Strategic |

Poker |

#### What is the meaning of first passage time?

For a given scale r, it is defined as the time required by the animals to pass through a circle of radius r. The mean first passage time scales proportionately to the square of the radius of the circle for an uncorrelated random walk (Johnson et al. 1992).

**What determines the first passage time for a random walk?**

The mean first passage time scales proportionately to the square of the radius of the circle for an uncorrelated random walk (Johnson et al. 1992).

**What is the value of FPT for a random walk?**

Under the hypothesis of a random walk, a should be equal to 2 (higher for impeded diffusion, and lower for facilitated diffusion). Note however, that the value of a converges to 2 only for large values of radius. Another use of the FPT was proposed that, instead of computing the mean of FPT, use the variance of the log (FPT).