Stochastic Processes

Question 1

Define a Poisson process

Answer 1

A Poisson process with rate lambda is a continuous time integer valued process Nt, t>=0 with the following properties.

1. N0 = 0
2. Nt has independent increments
3. Nt - Ns ~ Pois (lambda(t-s)) where t>s

Question 2

Weakly Stationary

Answer 2

a stochastic process Xt is said to be weakly stationary if E[Xt] = c where c is some constant for all t and the cov(Xt,Xt+k) depends on the lag k and not t

Question 3

Stochastic Process

Answer 3

model for a time-dependent random phenomenon, a collection of random variables { Xt: tEJ }, one for each time t in time set J. The time set/state space may be discrete and continuous.

Question 4

Sample paths

Answer 4

A joint realisation of the random variables Xt for all t in

Question 5

Counting Process

Answer 5

stochastic process, X, in discrete or counting time, whose state S is a collection of natural numbers {0,1,2…} with the property that Xt is a non-decreasing function of t

Question 6

Markov Property

Answer 6

probabilities for the future values of a process are dependent only on the recent value.

*know how to write mathematically