Question 23 - Week 2 Practice

Question 23

00:00Est. 2 min

Marks:+2.000.00

Passage

A system can be in one of

3

possible states at a given time. At the next instant, it changes its state as represented pictorially in the diagram. The number beside an arrow shows the transition probabilities from the beginning state of the arrow to the ending state. No arrow from state

2

to state

1

indicates that direct transition is not possible, so the transition probability is

0

. The probability of transition from a state to itself is

0

. The information in the diagram is represented by the matrix

P=\begin{bmatrix}0&\frac12&\frac12\\0&0&1\\\frac12&\frac12&0\end{bmatrix}

, where the

ij

-th entry of

P

denotes the probability of transition from state

i

to state

j

. Let the initial distribution vector be

X_0=\begin{bmatrix}X_0^1&X_0^2&X_0^3\end{bmatrix}^T

. For any positive integer

n

, the distribution vector at

t=n

is denoted by

X_n

and is given by

P^TX_{n-1}=X_n

. Answer the given subquestions.

Question

Suppose at

t=2

the distribution vector is

X_2=\begin{bmatrix}\frac13&\frac12&\frac23\end{bmatrix}^T

. Which of the following options are true?

Comprehension

Medium Difficulty

16 October 2022