Markov Chains and Stationary Distributions - Stochastic Processes - Statistics Previous Year Questions

1

2019 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2019

Consider a discrete time Markov chain on the state space {1,2,3} with one-step transition probability matrix \(\begin{pmatrix} 1 & 2 & 3 \\ 1 & 0.7 & 0.3 & 0 \\ 2 & 0 & 0.6 & 0.4 \\ 3 & 0 & 0 & 1 \end{pmatrix}\). Which of the following statements is true?

A

States 1, 3 are recurrent and state 2 is transient.

B

State 3 is recurrent and states 1, 2 are transient.

C

States 1, 2, 3 are recurrent.

D

States 1, 2 are recurrent and state 3 is transient.

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2

2019 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2019

Let \( \{X_n\}_{n \geq 0} \) be a discrete time Markov chain on the state space \( \{1,2,3\} \) with one-step transition probability matrix

	1	2	3
1	0.4	0.3	0.3
2	0.5	0.2	0.3
3	0.2	0.4	0.4

and initial distribution \( P(X_0 = 1) = 0.5, P(X_0 = 2) = 0.2, P(X_0 = 3) = 0.3 \). Then \( P(X_1 = 2, X_2 = 3, X_3 = 1) \) (rounded off to three decimal places) is equal to ...

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3

2019 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2019

Consider a discrete time Markov chain on the state space {1,2} with one-step transition probability matrix \[ P = \begin{pmatrix} 0.2 & 0.8 \\ 0.3 & 0.7 \end{pmatrix} \] Then \(\lim_{n\to\infty} P^n\) is

A

\(\begin{pmatrix} \frac{3}{11} & \frac{8}{11} \\ \frac{3}{11} & \frac{8}{11} \end{pmatrix}\)

B

\(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

C

\(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)

D

\(\begin{pmatrix} \frac{8}{11} & \frac{3}{11} \\ \frac{8}{11} & \frac{3}{11} \end{pmatrix}\)

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4

2019 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2019

Consider a discrete time Markov chain on the state space \( \{1,2,3\} \) with one-step transition probability matrix

	1	2	3
1	0	0.2	0.8
2	0.5	0	0.5
3	0.6	0.4	0

. Then the period of the Markov chain is ...

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5

2020 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2020

Let \( \{X_n\}_{n \geq 0} \) be a homogeneous Markov chain with state space {0,1} and one-step transition probability matrix
\[ P = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{2}{3} & \frac{1}{3} \end{bmatrix} \]. If \( P(X_0 = 0) = \frac{1}{3} \), then
\( 27 \times E(X_2) = \) ___________ (correct up to two decimal places).

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6

2020 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2020

Let $\{X_n\}_{n \geq 0}$ be a homogeneous Markov chain whose state space is $\{0,1,2\}$ and whose one-step transition probability matrix is $P = \begin{bmatrix} 0 & 1 & 0 \\ 0.3 & 0 & 0.7 \\ 0 & 1 & 0 \end{bmatrix}$. Then $\lim_{n \to \infty} P(X_{2n} = 2 \mid X_0 = 2) =$ ______________ (correct up to one decimal place).

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7

2020 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2020

Consider a homogeneous Markov chain $\{X_n\}_{n \geq 0}$ with state space $\{0,1,2,3\}$ and one-step transition probability matrix $P = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{4} & \frac{3}{4} \\ 0 & 0 & 0 & 1 \end{bmatrix}$. Assume that $P(X_0 = 1) = 1$. Let $p$ be the probability that state 0 will be visited before state 3. Then $6 \times p =$ ______________

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8

2021 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2021

Let \(\{X_n\}_{n \ge 0}\) be a time-homogeneous discrete time Markov chain with state space \(\{0,1\}\) and transition probability matrix \[\begin{bmatrix} 0.25 & 0.75 \\ 0.75 & 0.25 \end{bmatrix}.\] If \(P(X_0 = 0) = P(X_0 = 1) = 0.5\), then \[\sum_{k=1}^{100} E[(X_{2k})^{2k}]\] equals __________

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9

2021 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2021

Let {X_n}_{n≥0} be a time-homogeneous discrete time Markov chain with either finite or countable state space S. Then which one of the following statements is true?

A

There is at least one recurrent state

B

If there is an absorbing state, then there exists at least one stationary distribution

C

If all the states are positive recurrent, then there exists a unique stationary distribution

D

If {X_n}_{n≥0} is irreducible, S = {1, 2} and [π1, π2] is a stationary distribution, then lim_{n→∞} P(X_n = i | X_0 = i) = π_i for i = 1, 2

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10

2022 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2022

Let \( \{X(t)\\}_{t \geq 0} \) be a linear pure death process with death rate \( \mu_i = 5i, \; i = 0, 1, ..., N, \; N \geq 1 \). Suppose that \( p_i(t) = P(X(t) = i) \). Then the system of forward Kolmogorov’s equations is

A

\(\frac{dp_i(t)}{dt} = 5(i + 1)p_{i+1}(t) + 5i p_i(t) \) and \(\frac{dp_N(t)}{dt} = 5N p_N(t) \) for \(i = 0, 1, 2, ..., N - 1 \) with initial conditions \(p_i(0) = 0 \) for \(i \neq N \), and \(p_N(0) = 1 \)

B

\(\frac{dp_i(t)}{dt} = 5(i + 1)p_{i+1}(t) - 5i p_i(t) \) and \(\frac{dp_N(t)}{dt} = -5N p_N(t) \) for \(i = 0, 1, 2, ..., N - 1 \) with initial conditions \(p_i(0) = 0 \) for \(i \neq N \), and \(p_N(0) = 1 \)

C

\(\frac{dp_i(t)}{dt} = 5(i + 1)p_{i+1}(t) + 5i p_i(t) \) and \(\frac{dp_N(t)}{dt} = 5N p_N(t) \) for \(i = 0, 1, 2, ..., N - 1 \) with initial conditions \(p_i(0) = 1 \) for \(i \neq N \), and \(p_N(0) = 0 \)

D

\(\frac{dp_i(t)}{dt} = 5(i + 1)p_{i+1}(t) - 5i p_i(t) \) and \(\frac{dp_N(t)}{dt} = -5N p_N(t) \) for \(i = 0, 1, 2, ..., N - 1 \) with initial conditions \(p_i(0) = 1 \) for \(i \neq N \), and \(p_N(0) = 0 \)

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11

2022 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2022

Consider the following transition matrices \(P_1\) and \(P_2\) of two Markov chains:
\(P_1 = \begin{bmatrix} 1 & 0 & 0 \\ 1/3 & 1/2 & 1/6 \\ 0 & 0 & 1 \end{bmatrix}\) and \(P_2 = \begin{bmatrix} 1/6 & 1/3 & 1/2 \\ 1/4 & 0 & 3/4 \\ 0 & 1 & 0 \end{bmatrix}\).
Then which one of the following statements is true?

A

Both \(P_1\) and \(P_2\) have unique stationary distributions

B

\(P_1\) has a unique stationary distribution, but \(P_2\) has infinitely many stationary distributions

C

\(P_1\) has infinitely many stationary distributions, but \(P_2\) has a unique stationary distribution

D

Neither \(P_1\) nor \(P_2\) has unique stationary distribution

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12

2022 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2022

A company sometimes stops payments of quarterly dividends. If the company pays the quarterly dividend, the probability that the next one will be paid is 0.7. If the company stops the quarterly dividend, the probability that the next quarterly dividend will not be paid is 0.5. Then the probability (rounded off to three decimal places) that the company will not pay quarterly dividend in the long run is ________

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13

2022 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2022

In a laboratory experiment, the behavior of cats are studied for a particular food preference between two foods A and B. For an experiment, 70% of the cats that had food A will prefer food A, and 50% of the cats that had food B will prefer food A. The experiment is repeated under identical conditions. If 40% of the cats had food A in the first experiment, then the percentage (rounded off to one decimal place) of cats those will prefer food A in the third experiment, is ______

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14

2023 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2023

Let \( \{X_n\}_{n \geq 1} \) be a Markov chain with state space \( \{1, 2, 3\} \) and transition probability matrix
\[ \begin{bmatrix} \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \]
Then \( P(X_2 = 1 | X_1 = 1, X_3 = 2) \) (rounded off to two decimal places) equals ______________

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15

2023 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2023

Consider a birth-death process on the state space \(\{0, 1, 2, 3\}\). The birth rates are given by \(\lambda_0 = 1, \lambda_1 = 1, \lambda_2 = 2\) and \(\lambda_3 = 0\). The death rates are given by \(\mu_0 = 0, \mu_1 = 1, \mu_2 = 1\) and \(\mu_3 = 1\). If \([\pi_0, \pi_1, \pi_2, \pi_3]\) is the unique stationary distribution, then \(\pi_0 + 2\pi_1 + 3\pi_2 + 4\pi_3\) (rounded off to two decimal places) equals ______________

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16

2024 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2024

Let {X_n}_{n≥1} be a time homogeneous discrete time Markov chain with state space {0, 1, 2} and transition probability matrix \[ \begin{pmatrix} \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{pmatrix} \] Which of the following statements is/are true?

A

0 and 1 are recurrent states

B

2 is a transient state

C

Markov chain has a unique stationary distribution

D

Markov chain is irreducible

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17

2024 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2024

Let \(\{X_n\}_{n \geq 1}\) be a time homogeneous discrete time Markov chain with state space \(\{0, 1, 2\}\) and transition probability matrix \(\begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \end{bmatrix}\). If \(P(X_0 = 0) = P(X_0 = 1) = \frac{1}{4}\), then \(32 E(X_2)\) equals ______________ (in integer).

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18

2025 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2025

Consider a Markov chain \(\{X_n: n = 1, 2, ...\}\) with state space \(S = \{1, 2, 3\}\) and transition probability matrix \(P = \begin{pmatrix} 0 & 1/2 & 1/2 \\ 1/3 & 0 & 2/3 \\ 2/5 & 3/5 & 0 \end{pmatrix}\). Define \(\pi = \left(\frac{18}{67}, \frac{24}{67}, \frac{25}{67}\right)\). Which of the following options is/are correct?

A

\(\pi\) is a stationary distribution of \(P\)

B

\(\pi^T\) is an eigen vector of \(P^T\)

C

\(\Pr(X_3 = 1 \mid X_1 = 1) = \frac{11}{30}\)

D

At least one state is transient

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19

2026 · Statistics · Stochastic Processes · Markov Chains and Stationary Distributions

Statistics (ST) 2026

Let \(\{X_n: n \ge 0\}\) be a homogeneous Markov chain with state space \(S = \{1, 2, ..., 7\}\) and transition probability matrix
\[P = \begin{pmatrix} \frac{1}{3} & 0 & \frac{2}{3} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{3} & 0 & \frac{1}{3} & 0 & \frac{1}{3} & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 \\ \frac{1}{2} & 0 & 0 & 0 & \frac{1}{4} & \frac{1}{4} & 0 \\ 0 & \frac{2}{3} & 0 & \frac{1}{6} & 0 & \frac{1}{6} & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & 0 & \frac{1}{3} \end{pmatrix}.\]
Then which of the following statements is correct?

A

2 is a transient state

B

3 is a transient state

C

4 is a transient state

D

5 is a transient state

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