Stochastic Processes

Question 1

(A∪B)’

Answer 1

A’∩B’

Question 2

(A∩B)’

Answer 2

A’∪B’

Question 3

(A∪B)∩C

Answer 3

(A∩C)∪(B∩C)

Question 4

(A∩B)∪C

Answer 4

(A∪C)∩(B∪C)

Question 5

Ratio test

Answer 5

Σa_n
L=lim n→∞ |a_n+1| / |a_n|

L < 1 ABSOLUTELY CONVERGENT
L > 1 DIVERGENT
L = 1 UNKNOWN

Question 6

RV ‘X’ is normally distributed, notation

Answer 6

X ~ N(μ,σ²)

Question 7

Var[aX]

Answer 7

= a²Var[X]

Question 8

λ_eff

Answer 8

λ_eff = Σ(n=0) λ_n ⋅ P_n

Question 9

What should you be careful of when drawing the steady state Diagram for a M/M/2 model?

Answer 9

μ_1

The rate of μ_1 will be less that μ_2 since one server will be idle.

Question 10

Accessible states

Answer 10

State j is accessible from state i if

∃n>0 : (T^n)_ij > 0

Question 11

Communicating states

Answer 11

States i and j communicate with each other if they are accessible to each other

Question 12

Subchain

Answer 12

Equivalence classes of communicating states

Question 13

Irreducible chain

Answer 13

A chain that cannot be split into further subchains

Question 14

Absorbing state

Answer 14

State i is absorbing if T_ii = 1

Question 15

Periodic state

Answer 15

State i has minimal period k ≥ 2 if
P( X_n = i | X_0 = i ) = 0 IF n =/= k, 2k, 3k, …
P( X_n = i | X_0 = i ) > 0 IF n = k, 2k, 3k, …

Question 16

Aperiodic state

Answer 16

State i is not periodic, then it is aperiodic.
It is possible to find times n1, n2 s.t.
(T^n1)_ii, (T^n2)_ii > 0 
AND
gcd(n1,n2) = 1

Question 17

f_i

Answer 17

Probability of eventual return to state i

Question 18

f_i ^(n)

Answer 18

Probability of first return to state i at time n

Question 19

How to determine if a state is transient or recurrent?

Answer 19

f_i = 1 RECURRENT

f_i < 1 TRANSIENT

Question 20

How to determine if a state is positively or null recurrent?

Answer 20

μ_i < ∞ POSITIVELY RECURRENT

μ_i = ∞ NULL RECURRENT

Question 21

Ergodic chain

Answer 21

All states are
Aperiodic &
Positively recurrent

Question 22

θ_i in Simple Random walks

Answer 22

Probability of entering a state in W given that we start in state i∉A

Question 23

Z = X1 + X2 + … + XN

PGF of Z

Answer 23

G_Z(θ) = G_N( G_X(θ) )

Question 24

When is e=1, and e<1?

Answer 24

(1) e = 1 when μ_X ≤ 1

(2) e < 1 when μ_X > 1

Question 25

What conditions in Poisson processes question requires the use of the binomial?

Answer 25

Poisson process n events have happened in time T X ~ the number of events by time t X ~ B( n , t/T )

Question 26

When can you also use the solution to P( N(t) = k )? | P( N(t) ≤ k ) or P( N(t) ≥ k )

Answer 26

P( N(t) ≤ k )

Question 27

P( S_k > t ) | P( S_k ≤ t )

Answer 27

P( S_k > t ) = P( N(t) < k ) = P( N(t) ≤ k-1 ) | P( S_k ≤ t ) = P( N(t) ≥ k ) = 1 - P( N(t) ≤ k-1 )

Question 28

Cov( X, Y )

Answer 28

Cov( X, Y ) = E[XY] - E[X]E[Y]

Question 29

Tower Law

Answer 29

E[E[ X|Y ]] = E[X] | outer expectation w.r.t. y

Question 30

Poisson Process

Answer 30

The intervals of time between successive events are independent and follow an exponential distribution (inter-event times are iid)

Question 31

X ~ Po( λ ) | What is E[X]?

Answer 31

E[X] = λ | Also Var[X] = λ

Question 32

T ~ Exp( λ ) | What is E[T]?

Answer 32

E[T] = 1/λ

Question 33

Define MGFs

Answer 33

M_X(t) = E[ exp^(tX) ]

Question 34

Define PGFs

Answer 34

G_X(θ) = E[ θ^X ]