CSIR NET Maths previous year question papers

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?

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?

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 $Y_{j} = 1$ if $X_{j} \le \sqrt{5}$, 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

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?

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?

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?

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?

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?

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?

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?

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

$\lim(n\to\infty) \int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1} (x_{1}^{2} + \cdots + x_{n}^{2})/(x_{1} + \cdots + x_{n})$ dx$_{1} \dots$ dx$_{n}$ equals

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

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

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

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 $\alpha$ is

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

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?

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?

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed (i.i.d.) random variables such that for $x = 3, 4, \dots, P(X_{1} = \pm{}x) = 1/(2$cx$^{2}\ln(x))$, where $c = \sum(x=3..\infty) 1/(x^{2}\ln(x))$. Then, which of the following statements are true?

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?

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?

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?

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?

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?

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?