Probability, Statistics & Linear Programming

Limit Theorems and Markov Chains

29 previous-year questions →every option and the correct answer, free Practice this topic →one question at a time, no clock, check as you go

See where the marks are across every paper →

1. Modes of convergence, WLLN, SLLN, CLT

Read this as a chapter →

Lec-39 Modes of convergence

NPTEL · Measure and Integration

The implication diagram, done measure-theoretically.

Lec-40 Convergence in Measure

NPTEL · Measure and Integration

Lec-04 Problems in Sequences of Random Variables

NPTEL · Stochastic Processes

Worked problems on LLN and CLT-type limits.

Modes of convergence and the limit theorems free

In plain words

There are several ways to settle down, in a strict hierarchy. The laws of large numbers need the mean to exist, and the central limit theorem needs finite variance rather than normal data.

The diagram

   L^p  ──────▶  probability  ──────▶  distribution
                     ▲
almost sure ─────────┘

No other arrows hold in general. Partial converses:

convergence in probability ⇒ an a.s. convergent subsequence;
convergence in distribution to a constant ⇒ convergence in probability;
probability + uniform integrability $\Rightarrow L^{1}$ ; $- \sum P($ | $X_{n} - X$ | $> \varepsilon) < \infty \Rightarrow a.s. ($ Borel–Cantelli).

The counterexamples

Missing arrow	Example
probability ⇏ a.s.	typewriter: indicators of $[k/2^{n}, (k+1)/2^{n}]$ sweeping [0,1]
a.s. ⇏ $L^{1}$	$X_{n} = n\cdot1_{(0,1/n)}$ : $\to 0 a.s., E[X_{n}] = 1$
probability ⇏ $L^{1}$	same
distribution ⇏ probability	X ~ $N(0,1), X_{n} = -X$ for all n
$L^{1}$ ⇏ a.s.	typewriter again

Laws of large numbers

WLLN (in probability) needs finite mean; SLLN (a.s.) also holds with finite mean (Kolmogorov).
Fails without a mean: the Cauchy sample mean has the same Cauchy distribution for every n.

Central limit theorem

$\sqrt{n}({\bar{X}} - \mu)/\sigma \to N(0,1)$ whenever $\sigma^{2} < \infty$ . Consequences examined: $- P(S_{n} \le n\mu) \to$ ½, so intervals with the mean as an endpoint have limiting probability ½, while intervals containing $\mu$ in the interior have probability → 1.

Delta method: $\sqrt{n}(g({\bar{X}}) - g(\mu)) \to N(0, \sigma^{2}g'(\mu)^{2})$ .
Slutsky: $X_{n} \Rightarrow X$ and $Y_{n} \to c$ in probability $\Rightarrow X_{n} + Y_{n} \Rightarrow X + c, X_{n}Y_{n} \Rightarrow$ cX.

Key takeaways

a.s. and L^p both imply probability, which implies distribution; nothing else.
Typewriter kills "probability ⇒ a.s."; n· $1_{(0,1/n)}$ kills " $a.s. \Rightarrow L^{1}$ ".
CLT at the boundary gives ½ — a favourite Part-C trick.

The Central Limit Theorem, watchedinteractivefree

Exam focus: Know the implication diagram and one counterexample per missing arrow. The typewriter sequence and the moving-bump are the two you need.

Practice

Part C2025 Juneshow ▾

Let

X_{1}, X_{2}, \dots

be a sequence of independent and identically distributed random variables with

E(X_{1}) = 0, E(X_{1}^{2}) = 1, E(X_{1}^{3}) = 0, E(X_{1}^{4}) = 3

. Let

S_{n} = \sum_{i}

₌

_{1}^{n} X_{i}, T_{n} = \sum_{i}

₌

_{1}^{n} X_{i}^{2}, U_{n} = \sum_{i}

₌

_{1}^{n} X_{i}^{3}

and

V_{n} = \sum_{i}

₌

_{1}^{n} X_{i}^{4}

. Then, which of the following statements are true?

A. $S_{n}/\sqrt{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)✓ Correct
B. $(T_{n} - n)/\sqrt{3n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)
C. $\sqrt{n} S_{n}/T_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)✓ Correct
D. $(T_{n} - n)/\sqrt{V}_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

Let

X_{1}, X_{2}, \dots

be a sequence of independent and identically distributed random variables having Binomial(3, 1/4) distribution. For j=1,2,…, define

Y_{j} = 1

X_{j} \le \sqrt{5}

, and 0 otherwise. If

(1/n)\sum_{j=1}^n

Y_{j}^{2}

converges almost surely to a constant c as

n \to \infty

, then the value of c equals

A.1/64
B.37/64
C.63/64✓ Correct
D.27/32

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Suppose that a sequence of random variables {

X_{n}

}

_{n}\ge_{1}

and the random variable X are defined on the same probability space. Then which of the following statements are true?

A. $X_{n}$ converges to X almost surely as $n \to \infty$ implies that $X_{n}$ converges to X in probability as $n \to \infty$ .✓ Correct
B. $X_{n}$ converges to X in probability as $n \to \infty$ implies that $X_{n}$ converges to X almost surely as $n \to \infty$ .
C.If $\sum(n=1..\infty)$ ℙ[| $X_{n} - X$ | $> \delta] < \infty$ for all $\delta > 0$ , then $X_{n}$ converges to X almost surely as $n \to \infty$ .✓ Correct
D.If $X_{n}$ converges to X in distribution as $n \to \infty$ , and X is a constant with probability 1, then $X_{n}$ converges to X in probability as $n \to \infty$ .✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let {(X_k, Y_k)}(k≥1) be a sequence of independent and identically distributed (i.i.d.) random vectors with common joint probability density function f(x, y) = e^(−y) if

0 < x < y < \infty

, and 0 otherwise. For n = 1, 2, 3, …, let

N_{n}

be a random variable denoting the number of elements in the set {k : k = 1, 2, …, n; Y_k ≥ 2}. Then, which of the following statements are true?

A. $N_{n}/(3n)$ converges to e⁻ $^{2}$ with probability one.✓ Correct
B. $N_{n}$ converges to e⁻ $^{2}$ in probability.
C. $N_{n}/n$ converges to 3e⁻ $^{2}$ in distribution.✓ Correct
D. $(N_{n} - 3$ ne⁻ $^{2})/\sqrt{3n}$ converges in distribution to a normal random variable with mean zero and variance e⁻ $^{2}(1 - 3e$ ⁻ $^{2})$ .✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

U_{1}, U_{2}, \dots

be a sequence of independent and identically distributed (i.i.d.) U(0, 1) random variables. Let

G_{n} = (\prod(i=1..n) U_{i})^(1/n)

be the geometric mean of

U_{1}, U_{2}, \dots, U_{n}

for

n \in \mathbb{N}

. Let

X_{1}

and

X_{2}

be degenerate random variables such that

P(X_{1} = 0) = 1

and

P(X_{2} = 1/e) = 1

. Then, which of the following statements are true?

A. $G_{n}$ converges in r-th mean to $X_{1}$ as $n \to \infty$ , for any r > 0
B. $G_{n}$ converges in probability to $X_{1}$ as $n \to \infty$
C. $G_{n}$ converges in distribution to $X_{2}$ as $n \to \infty$ ✓ Correct
D. $G_{n}$ converges in r-th mean to $X_{2}$ as $n \to \infty$ , for any r > 0✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

X_{1}, X_{2}, \dots, X_{n}

be independent and identically distributed (i.i.d.) random variables such that for

x = 3, 4, \dots, P(X_{1} = \pm{}x) = 1/(2

^{2}\ln(x))

, where

c = \sum(x=3..\infty) 1/(x^{2}\ln(x))

. Then, which of the following statements are true?

A.E(| $X_{1}$ | $) = \infty$ ✓ Correct
B. $(1/n)\sum(i=1..n)X_{i} \to 0$ in probability as $n \to \infty$ ✓ Correct
C.E(1/| $X_{1}$ | $) < \infty$ ✓ Correct
D. $E(X_{1}^{2}) = \infty$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

\lim(n\to\infty) \int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1} (x_{1}^{2} + \cdots + x_{n}^{2})/(x_{1} + \cdots + x_{n})

_{1} \dots

_{n}

equals

A.1
B.1/2
C.2
D.2/3✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Suppose that {

X_{n}

}

(n\in\mathbb{N})

is a sequence of independent and identically distributed (i.i.d.) random variables with the common probability density function

f(x) = 1/(\pi(1 + x^{2})), x \in \mathbb{R}

. Then which of the following statements is true?

A. $X_{1}$ and $(X_{1} + X_{2})/\sqrt{2}$ have the same distribution.
B. $(1/n)\sum(i=1..n)X_{i}$ converges to 0 in probability, as $n \to \infty$ .
C.Median of { $X_{1}, X_{2}, \dots, X(2n+1)$ } converges to 0 in probability, as $n \to \infty$ .✓ Correct
D.E(| $X_{1}$ | $^(3/4)) = \infty$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let {

X_{n}

}

_{n}\ge_{1}

be a sequence of independent and identically distributed random variables with

E(X_{1}) = 0

and

Var(X_{1}) = 1

. Which of the following statements are true?

A. $\lim(n\to\infty) P((\sqrt{n}\cdot\sum_{i}$ ₌ $_{1}^{n}X_{i})/(\sum_{i}$ ₌ $_{1}^{n}X_{i}^{2}) \le 0) = 1/2$ ✓ Correct
B. $(\sum_{i}$ ₌ $_{1}^{n}X_{i})/(\sum_{i}$ ₌ $_{1}^{n}X_{i}^{2})$ converges in probability to 0 as $n \to \infty$ ✓ Correct
C. $(1/n)\sum_{i}$ ₌ $_{1}^{n}X_{i}^{2}$ converges in probability to 1 as $n \to \infty$ ✓ Correct
D. $\lim(n\to\infty) P((\sum_{i}$ ₌ $_{1}^{n}X_{i})/\sqrt{n} \le 0) = 1/2$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

X_{1}, X_{2} \dots

be a sequence of independent random variables with

X_{i}

following N(0, 1 + 1/i) distribution for all

i \in \mathbb{N}

. Let X be N(0, 1)-random variable independent of {

X_{n} : n \ge 1

}. Then, which of the following statements are true?

A. $X_{n}$ converges to X in probability as $n \to \infty$
B. $X_{n}$ converges to X in distribution as $n \to \infty$ ✓ Correct
C. $X_{n} - X$ converges to Z in distribution as $n \to \infty$ , where Z follows the distribution N(0, 2).✓ Correct
D.X/| $X_{n}$ | converges to M in distribution as $n \to \infty$ , where M follows standard Cauchy distribution.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾

Let

X_{1}, X_{2}, \dots

be i.i.d. with CDF F(x) = 0 for x < 5 and 1 −

e^{5-x}

for x ≥ 5. Define

Y_{n} = \min

{

X_{1}, \dots, X_{n}

} and

Z_{n} = \sqrt{n} (Y_{n} - 5)

, and let Z be standard normal. Which one of the following statements is true?

A. $\lim P(1/2 < Y_{n} < 3/2) = 1$
B. $Y_{n} \to 5$ in probability as $n \to \infty$ ✓ Correct
C. $Z_{n} \to Z$ in distribution as $n \to \infty$
D. $\lim P(1 < Z_{n} < 2) = \Phi(2) - \Phi(1)$

Solution

$Y_{n} - 5$ ~ Exp(n), so $Y_{n} \to 5$ in probability while $\sqrt{n}(Y_{n} - 5) \to 0$ in probability (the right scaling is n, not $\sqrt{n})$ .

Save for spaced revision (free account)

Part C2023 Juneshow ▾

Suppose

X_{1}, X_{2}, \dots

are independent and identically distributed N(0,1) random variables and

Y_{n} = X_{1}^{4} + X_{2}^{4} + \cdots + X_{n}^{4}

. Which of the following probabilities converge to 1/2 as

n \to \infty

A.P{ $Y_{n} \in [0, 2n]$ }
B.P{ $Y_{n} \in [n, 3n]$ }✓ Correct
C.P{ $Y_{n} \in [2n, 4n]$ }
D.P{ $Y_{n} \in [3n, 5n]$ }✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

X_{1}, X_{2}, \dots

and X be random variables on a common probability space. Which of the following implications hold in general?

A. $X_{n} \to X$ almost surely ⟹ $X_{n} \to X$ in probability.✓ Correct
B. $X_{n} \to X$ in probability ⟹ $X_{n} \to X$ almost surely.
C. $X_{n} \to X$ in probability ⟹ $X_{n} \to X$ in distribution.✓ Correct
D. $X_{n} \to X$ in distribution ⟹ $X_{n} \to X$ in probability.

Solution

The standard hierarchy is: almost sure ⟹ in probability ⟹ in distribution, and no arrow reverses in general.

(2) Fails for the sliding-block ('typewriter') sequence of indicators on [0,1], which tends to 0 in probability while diverging at every point.

(4) Fails whenever the limit is not degenerate: take X ~ N(0,1) and $X_{n} = -X$ for all n.

Then $X_{n}$ has the same distribution as X for every n, so $X_{n} \to X$ in distribution, yet | $X_{n} - X$ | = 2|X| does not tend to 0 in probability.

Convergence in distribution does imply convergence in probability when the limit is a constant.

Save for spaced revision (free account)

Part Cshow ▾

Let

X_{1}, X_{2}, \dots

be i.i.d. with E|

X_{1}

< \infty, E X_{1} = \mu

, and

{\bar{S}}_{n} = (X_{1} + \cdots + X_{n})/n

. Which of the following are true?

A. ${\bar{S}}_{n} \to \mu$ almost surely.✓ Correct
B.If additionally $0 < Var X_{1} = \sigma^{2} < \infty$ , then $\sqrt{n}({\bar{S}}_{n} - \mu)/\sigma$ converges in distribution to N(0, 1).✓ Correct
C.Under the assumptions of option $2, {\bar{S}}_{n} - \mu = O_p(n^{-1/2})$ , so ${\bar{S}}_{n} \to \mu$ in probability.✓ Correct
D.The strong law of large numbers requires finite variance.

Solution

(1) Kolmogorov's strong law needs only a finite first moment.

(2) is the classical CLT.

(3) tightness of $\sqrt{n}({\bar{S}}_{n} - \mu)$ forces ${\bar{S}}_{n} - \mu \to 0$ in probability.

(4) fails — finite mean suffices for the SLLN; finite variance is a hypothesis of the CLT, not of the strong law.

Save for spaced revision (free account)

Part Bshow ▾

X_{n} \to X

in probability, then necessarily

A. $X_{n} \to X$ almost surely
B. $X_{n} \to X$ in distribution✓ Correct
C.E| $X_{n} - X$ | → 0
D. $X_{n} \to X$ in $L^{2}$

The worked solution and the per-option analysis are in Trap Analysis.

2. Markov chains: classification of states, stationary distributions

Read this as a chapter →

Lec-01 Introduction, Definition and Transition Probability Matrix

NPTEL · Stochastic Processes

Lec-03 Classification of States and Limiting Distributions

NPTEL · Stochastic Processes

Recurrence, transience and periodicity as class properties.

Lec-04 Limiting and Stationary Distributions

NPTEL · Stochastic Processes

Where aperiodicity is needed for convergence.

Markov chains: classification and stationary behaviour free

In plain words

The chain forgets everything except where it is now. Reachability makes the long-run distribution unique, positive recurrence makes it exist, and aperiodicity makes the chain actually converge to it.

Classification

Communicating classes; a chain is irreducible if there is one class.
A state is recurrent if return is certain $(\sum{}p_{nii} = \infty)$ , else transient.
Positive recurrent if the mean return time $m_{ii}$ is finite; else null recurrent.
Period = gcd{ $n : p_{nii} > 0$ }; aperiodic if 1.
Recurrence, transience and period are class properties.
A finite irreducible chain is always positive recurrent.

Stationary distributions

$\pi = \pi{}P$ with $\sum\pi = 1$ .

Irreducible + positive recurrent ⇒ a unique stationary $\pi$ with $\pi_{i} = 1/m_{ii}$ .
Add aperiodic $\Rightarrow p_{nij} \to \pi_{j} ($ the chain converges to $\pi$ from any start).
Periodic chains have a unique $\pi$ but no convergence — the standard trap.
Reducible chains can have infinitely many stationary distributions.

Standard chains

Chain	Behaviour
Simple random walk on $\mathbb{Z}, p =$ ½	null recurrent (returns, but mean time infinite)
Same with p ≠ ½	transient
SRW on $\mathbb{Z}^{2}$	null recurrent; on $\mathbb{Z}$ ᵈ, d ≥ 3: transient (Pólya)
Finite irreducible	positive recurrent, unique $\pi$
Birth–death	reversible; $\pi$ from detailed balance

Detailed balance $\pi_{i}p_{ij} = \pi_{j}p_{ji} \Rightarrow \pi$ is stationary (reversible chain). Easiest route to $\pi$ when it applies.

Absorption

For transient states with fundamental matrix N = (I − Q)⁻ $^{1}$ : expected visits and absorption probabilities come from N and NR.

Key takeaways

Positive recurrent + irreducible ⇒ unique $\pi = 1/$ mean return time.
Aperiodicity is needed for convergence, not for existence of $\pi$ .
Pólya: the walk returns in dimensions 1 and 2, escapes from 3 up.

Why periodic chains never settleinteractive

This visual is part of Trap Analysis

A cycle with stationary distribution (⅓,⅓,⅓) that never converges to it — until one slider makes it aperiodic.

Unlock interactive visuals

Exam focus: Classify states (recurrent/transient, periodicity), then use irreducible + aperiodic + positive recurrent ⇒ unique stationary distribution with πⱼ = 1/mⱼⱼ.

Practice

Part C2025 Decembershow ▾

Consider an M/M/2 queuing system with the birth rate

\lambda=4

per minute, the death rate

\mu=1

per minute, and the total capacity of 3 customers (including the ones that are being served). Let

p_{0}

and

p_{3}

denote the long-run probabilities that the system will be empty (i.e. without customers) and will be blocked (i.e. full), respectively. Which of the following statements are true?

A. $p_{0}+p_{3} = 17/29$ ✓ Correct
B. $p_{0} > p_{3}$
C. $p_{3}/p_{0} = 2$
D. $p_{3}-p_{0} = 15/29$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

Let {

X_{n} : n \ge 0

} be any homogeneous Markov Chain on the state space S = {1,2,3,4} having the transition probability matrix

P = (p_{ij})

given by row 1 = (1/4, 0, 2/3, 1/12), row 2 = (0, 1, 0, 0), row 3 = (1/12, 0, 1/4, 2/3), row 4 = (2/3, 0, 1/12, 1/4). Which of the following statements about stationary distributions of any such Markov Chain is true?

A.There is no stationary distribution
B.Stationary distribution exists and is unique
C.There are exactly two stationary distributions
D.There are infinitely many stationary distributions✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

Consider the M/M/1 queue in which customers arrive according to a Poisson process with rate 3 and successive service times are independent exponential random variables having mean 1/9. Let

P_{n}

be the long run probability that there are exactly n customers in the system. Then, which of the following statements are true?

A. $P_{0} = 1/3$
B. $P_{1} = 2/9$ ✓ Correct
C.The average number of customers in the system is 1
D.The average amount of time that a customer spends in the system is 1/6✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

Suppose that the transition probability matrix of a homogeneous Markov chain with state space {1,2,3,4} is given by P, with row 1 = (1/4, 3/4, 0, 0), row 2 = (1, 0, 0, 0), row 3 = (1/8, 0, 7/8, 0), row 4 = (0, 0, 1/9, 8/9). Which of the following statements are true?

A.State 2 is a positive recurrent state✓ Correct
B.Mean recurrence time of state 1 is 7/4✓ Correct
C.State 4 is a transient state✓ Correct
D.State 3 is aperiodic and ergodic

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾

Let

Z_{1}, Z_{2}, \dots

be a sequence of independent and identically distributed random variables having discrete uniform distribution over {1, 2, …, 2024}. Let

Y_{n} = \sum_{i}

₌

_{1}^{n} Z_{i}, n \ge 2

. Further, let

X_{n}

be the remainder when

Y_{n}

is divided by 2025. Then, which of the following statements is true?

A. $lim_{n}\to\infty P(X_{n} = 0) = 1/2026$
B. $lim_{n}\to\infty P(X_{n} = 0) = 1/2025$ ✓ Correct
C. $lim_{n}\to\infty P(X_{n} = 0) = 1/2024$
D. $lim_{n}\to\infty P(X_{n} = 0) = 1/2023$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

Consider a Markov chain {

X_{n} : n \ge 1

} on state space {1, 2, 3, 4, 5} with the transition probability matrix whose rows are (0, 1/2, 1/2, 0, 0), (0, 0, 1, 0, 0), (0, 1/3, 0, 1/3, 1/3), (1, 0, 0, 0, 0) and (0, 0, 0, 0, 1). Then, which of the following statements are true?

A.Stationary distribution is (0, 0, 0, 0, 1).✓ Correct
B.State 5 is absorbing and recurrent.✓ Correct
C.All states are aperiodic.✓ Correct
D. $lim_{n}\to\infty p_{55}$ ⁽ $^{n}$ ⁾ = 1.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Consider an M/G/1 queuing system with arrival rate

\lambda = 1

and independent and identically distributed successive service times having probability density function g(x) = xe⁻ˣ if x > 0, and 0 otherwise. Define, for

i = 1, 2, \dots, I_{i} = 1

if the first transition is from i to i − 1, and

I_{i} = 0

if the first transition from i is i + 1. Then,

Var(I_{2})

equals

A.5/32
B.5/24
C.3/16✓ Correct
D.8/15

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Consider an M/M/3 queuing system with arrival rate

\lambda = 2

and service rate

\mu = 3/2

. Define, for

i = 1, 2, \dots, I_{i} = 1

if the first transition is from i to i + 1, and

I_{i} = 0

if the first transition from i is i − 1. Then,

Var(I_{4})

equals

A.33/169
B.34/169
C.35/169
D.36/169✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Consider the Markov chain with state space {0, 1, 2, 3, 4} and the transition probability matrix P = [[0.4, 0.3, 0.3, 0, 0], [0, 0.5, 0, 0.5, 0], [0.5, 0, 0.5, 0, 0], [0, 0.5, 0, 0.5, 0], [0, 0.3, 0, 0.3, 0.4]]. Then, which of the following statements are true?

A.The state space can be partitioned into exactly two equivalence classes.
B.States 0 and 2 are recurrent.
C.States 1 and 3 are recurrent.✓ Correct
D.State 4 is transient.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

In an examination question paper, all questions are 'True' or 'False' type. These are arranged in such a way that three-fourth of times a question with answer 'True' is followed by a question with answer 'True'. Also two-third of times a question with answer 'False' is followed by a question with answer 'False'. If the question paper has 100 questions, the approximate probability that the correct answer of the 100-th question is 'True', is

A.3/7
B.4/7✓ Correct
C.3/4
D.5/6

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Consider the Markov chain with state space {0, 1, 2} and the transition probability matrix P = [[0, 1/2, 1/2], [3/4, 0, 1/4], [3/4, 1/4, 0]]. Let P⁽

^{n}

⁾ = ((P⁽

^{n}

⁾

_{ij}))

denote the n-step transition probability matrix. Then, which of the following statements are true?

A.P⁽ $^{2}$ ⁾ $_{00} = 3/4$ ✓ Correct
B.P⁽ $^{3}$ ⁾ $_{10} = 39/64$ ✓ Correct
C.The stationary probability that the chain is in state 2 is 2/7.✓ Correct
D.State 1 is transient.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Consider a petrol pump which has a single petrol dispensing unit. Customers arrive there in accordance with a Poisson process having rate

\lambda = 1

minutes. An arriving customer enters the petrol pump only if there are two or less customers in the petrol pump, otherwise he/she leaves the petrol pump without taking the petrol (at any point of time a maximum of three customers are present in the petrol pump). Successive service times of the petrol dispensing unit are independent exponential random variables having mean 1/2 minutes. Let X denote the average number of customers in the petrol pump in the long run. Then E(X) is equal to

A.7/15
B.3/5
C.11/15✓ Correct
D.13/15

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Planes take off in a busy airport in accordance with the Poisson process with rate 60 planes per hour. 10% of these planes are cargo planes and 90% are passenger planes. Given that 10 cargo planes have taken off during one hour, what is the expected total number of planes that have taken off in that hour?

A.90
B.54
C.64✓ Correct
D.50

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Let {

X_{n}

| n ≥ 0} be a homogeneous Markov chain with state space S = {0, 1, 2, 3, 4} and transition probability matrix

	0	1	2	3	4
0	1/4	0	0	3/4	0
1	0	1	0	0	0
2	1/3	2/3	0	0	0
3	3/4	0	0	1/4	0
4	1/8	1/8	1/2	1/8	1/8

Let

\alpha

denote the probability that starting with state 4 the chain will eventually get absorbed in closed class {0, 3}. Then the value of

\alpha

A.6/21
B.11/21
C.8/21
D.10/21✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Transition probability matrix of a homogeneous Markov chain with states 0, 1, 2, 3 is P = [[1/4, 3/4, 0, 0], [1, 0, 0, 0], [2/3, 0, 1/3, 0], [0, 0, 2/5, 3/5]], where rows and columns are indexed 0, 1, 2, 3. Which of the following statements are true?

A.state 0 is positive recurrent✓ Correct
B.state 3 is transient✓ Correct
C.state 1 is aperiodic and positive recurrent✓ Correct
D.state 2 is aperiodic and null-recurrent

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

Consider an M/M/1 queue with arrival rate

\lambda = 15

per hour and service rate

\mu = 45

per hour. Let N(t) be the number of customers in the system at time t, and

T_{1}, T_{2}

the time a customer spends in the queue and in the system. Which of the following statements are true?

A. $lim_{t\to\infty}$ P(N(t) = 1) = 2/9✓ Correct
B. $P(T_{1} > 0) = 1/3$ ✓ Correct
C. $E(T_{1}) = 1/90$ ✓ Correct
D. $E(T_{2}) = 1/35$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

Suppose {X(t)} and {Y(t)} are two independent homogeneous Poisson processes with the same rate

\lambda = 2

. Let

W_{n}

ˣ and

W_{n}

ʸ be the waiting times for the n-th arrival in each process. Which of the following statements are true?

A. $P(W_{2}$ ˣ $< W_{3}$ ʸ) = 11/16✓ Correct
B. $P(W_{1}$ ˣ $< W_{1}$ ʸ) = 1/2✓ Correct
C. $P(W_{2}$ ˣ $< W_{3}$ ʸ) = 13/16
D. $P(W_{1}$ ˣ $< W_{1}$ ʸ) = 1/4

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Consider the Markov chain on {1, 2, 3} with transition matrix P = [[0, 1, 0], [0, 0, 1], [1, 0, 0]]. Which of the following are true?

A.The chain is irreducible.✓ Correct
B.Every state has period 3.✓ Correct
C. $\pi = (1/3, 1/3, 1/3)$ is a stationary distribution.✓ Correct
D. $P^{n}$ converges entrywise as $n \to \infty$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

A two-state Markov chain has transition matrix P = [[0.8, 0.2], [0.3, 0.7]]. Its stationary distribution is

A.(0.5, 0.5)
B.(0.6, 0.4)✓ Correct
C.(0.4, 0.6)
D.(0.8, 0.2)

The worked solution and the per-option analysis are in Trap Analysis.