CSIR NET Mathematical Science UNIT – 4 Syllabus

UNIT – 4 syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for UNIT – 4 in CSIR NET Mathematical Science, not a summary of it.

1. 1

   Descriptive Statistics and Exploratory Data Analysis

   8 topics

   • Sample Space and Discrete Probability
   • Independent Events and Bayes Theorem
   • Random Variables and Distribution Functions
     • Univariate Distributions
     • Multivariate Distributions
   • Expectation and Moments
   • Independent Random Variables
   • Marginal and Conditional Distributions
   • Characteristic Functions
   • Probability Inequalities
     • Tchebyshef's Inequality
     • Markov's Inequality
     • Jensen's Inequality
2. 2

   Convergence and Limit Theorems

   3 topics

   • Modes of Convergence
   • Weak and Strong Laws of Large Numbers
   • Central Limit Theorems
     • Independent and Identically Distributed (i.i.d.) Case
3. 3

   Markov Chains

   5 topics

   • Finite and Countable State Space
   • Classification of States
   • Limiting Behavior of n-step Transition Probabilities
   • Stationary Distribution
   • Poisson and Birth-and-Death Processes
4. 4

   Distributions and Sampling

   4 topics

   • Standard Discrete and Continuous Distributions
   • Sampling Distributions
   • Standard Errors and Asymptotic Distributions
   • Distribution of Order Statistics and Range
5. 5

   Estimation and Hypothesis Testing

   4 topics

   • Methods of Estimation
   • Properties of Estimators
   • Confidence Intervals
   • Tests of Hypotheses
     • Likelihood Ratio Tests
6. 6

   Regression Analysis

   2 topics

   • Simple and Multiple Linear Regression
     • Elementary Regression Diagnostics
   • Logistic Regression
7. 7

   Multivariate Analysis

   3 topics

   • Multivariate Normal and Wishart Distributions
     • Distribution of Quadratic Forms
   • Inference for Parameters
     • Partial and Multiple Correlation Coefficients
   • Data Reduction Techniques
     • Principal Component Analysis
     • Discriminant Analysis
     • Cluster Analysis
     • Canonical Correlation
8. 8

   Sampling Methods

   5 topics

   • Simple Random Sampling
   • Stratified Sampling
   • Systematic Sampling
   • Probability Proportional to Size Sampling
   • Ratio and Regression Methods
9. 9

   Experimental Design

   4 topics

   • Completely Randomized Designs
   • Randomized Block Designs
   • Latin-Square Designs
   • Connectedness and Orthogonality of Block Designs
     • BIBD (Balanced Incomplete Block Designs)
10. 10

    Factorial Experiments

    1 topic

    • 2K Factorial Experiments
      • Confounding and Construction
11. 11

    Reliability and Survival Analysis

    1 topic

    • Hazard Function and Failure Rates
      • Censoring and Life Testing
      • Series and Parallel Systems
12. 12

    Operations Research

    2 topics

    • Linear Programming
      • Simplex Methods
      • Duality
    • Elementary Queuing and Inventory Models
      • Steady-State Solutions of Markovian Queuing Models: M/M/1
      • M/M/1 with Limited Waiting Space
      • M/M/C
      • M/M/C with Limited Waiting Space
      • M/G/1

UNIT – 4 flashcards for CSIR NET Mathematical Science

24 of 51 cards from the UNIT – 4 deck — real questions with worked answers.

1. Define a sample space and an event in probability theory.

   The sample space $\Omega$ is the set of all possible outcomes of a random experiment. An event $A$ is any subset of the sample space, i.e. $A \subseteq \Omega$. An event occurs if the realized outcome belongs to $A$.

2. State the three Kolmogorov axioms of probability.

   For a probability measure $P$ on events: (1) Non-negativity: $P(A) \geq 0$ for all events $A$. (2) Normalization: $P(\Omega) = 1$. (3) Countable additivity: for pairwise disjoint events $A_1, A_2, \dots$, $P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i)$.

3. For a discrete sample space with $n$ equally likely outcomes, what is the probability of an event $A$?

   $$P(A) = \frac{|A|}{n} = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}.$$

4. State the inclusion-exclusion formula for $P(A \cup B)$ and for three events.

   For two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. For three events: $$P(A\cup B\cup C) = P(A)+P(B)+P(C) - P(A\cap B) - P(A\cap C) - P(B\cap C) + P(A\cap B\cap C).$$

5. Define the conditional probability of $A$ given $B$.

   For $P(B) > 0$, $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}.$$ It measures the probability of $A$ assuming $B$ has occurred.

6. When are two events $A$ and $B$ independent? Give the defining condition and the conditional form.

   $A$ and $B$ are independent iff $P(A \cap B) = P(A)\,P(B)$. Equivalently, if $P(B)>0$, $P(A\mid B) = P(A)$; the occurrence of $B$ gives no information about $A$.

7. Distinguish pairwise independence from mutual (joint) independence for events $A_1,\dots,A_n$.

   Pairwise independence requires $P(A_i \cap A_j) = P(A_i)P(A_j)$ for all $i \neq j$. Mutual independence requires that for every subset $S \subseteq \{1,\dots,n\}$, $P\left(\bigcap_{i\in S} A_i\right) = \prod_{i\in S} P(A_i)$. Mutual independence implies pairwise, but not conversely.

8. State the law of total probability for a partition $\{B_1,\dots,B_n\}$ of the sample space.

   If $B_1,\dots,B_n$ are disjoint with $\bigcup_i B_i = \Omega$ and $P(B_i)>0$, then $$P(A) = \sum_{i=1}^{n} P(A \mid B_i)\,P(B_i).$$

9. State Bayes' Theorem.

   For a partition $\{B_1,\dots,B_n\}$ with $P(B_i)>0$ and event $A$ with $P(A)>0$, $$P(B_k \mid A) = \frac{P(A \mid B_k)\,P(B_k)}{\sum_{i=1}^{n} P(A \mid B_i)\,P(B_i)}.$$

10. In Bayes' Theorem, name the terms $P(B_k)$, $P(A\mid B_k)$, and $P(B_k\mid A)$.

    $P(B_k)$ is the prior probability, $P(A\mid B_k)$ is the likelihood, and $P(B_k\mid A)$ is the posterior probability. The denominator $P(A)$ is the marginal (normalizing) probability of the evidence.

11. Define a random variable and its (cumulative) distribution function.

    A random variable $X$ is a measurable function $X:\Omega \to \mathbb{R}$. Its cumulative distribution function (CDF) is $F_X(x) = P(X \leq x)$ for $x \in \mathbb{R}$.

12. List the defining properties of a CDF $F(x)$.

    (1) Non-decreasing. (2) Right-continuous: $\lim_{h\to 0^+} F(x+h) = F(x)$. (3) $\lim_{x\to -\infty} F(x) = 0$ and $\lim_{x\to +\infty} F(x) = 1$. Also $P(a < X \leq b) = F(b) - F(a)$.

13. How are the pmf/pdf related to the CDF in the discrete and continuous cases?

    Discrete: $F(x) = \sum_{t \leq x} p(t)$, where $p(t) = P(X=t)$ is the probability mass function. Continuous: $F(x) = \int_{-\infty}^{x} f(t)\,dt$ and $f(x) = F'(x)$ where it exists, with $f$ the probability density function.

14. State the pmf, mean, and variance of the Binomial distribution $\text{Bin}(n,p)$.

    $$P(X=k) = \binom{n}{k} p^{k}(1-p)^{n-k}, \quad k=0,1,\dots,n.$$ Mean $E[X] = np$, variance $\operatorname{Var}(X) = np(1-p)$.

15. State the pmf, mean, and variance of the Poisson distribution with parameter $\lambda$.

    $$P(X=k) = \frac{e^{-\lambda}\lambda^{k}}{k!}, \quad k=0,1,2,\dots$$ Mean $E[X] = \lambda$ and variance $\operatorname{Var}(X) = \lambda$ (mean equals variance).

16. State the pmf, mean, and variance of the Geometric distribution (number of trials until first success).

    $$P(X=k) = (1-p)^{k-1} p, \quad k=1,2,\dots$$ Mean $E[X] = \frac{1}{p}$, variance $\operatorname{Var}(X) = \frac{1-p}{p^{2}}$. It is memoryless among discrete distributions.

17. State the pdf, mean, and variance of the Exponential distribution with rate $\lambda$.

    $$f(x) = \lambda e^{-\lambda x}, \quad x \geq 0.$$ Mean $E[X] = \frac{1}{\lambda}$, variance $\operatorname{Var}(X) = \frac{1}{\lambda^{2}}$. It is the unique continuous memoryless distribution.

18. State the pdf, mean, and variance of the Normal distribution $N(\mu,\sigma^{2})$.

    $$f(x) = \frac{1}{\sqrt{2\pi}\,\sigma} \exp\!\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right), \quad x \in \mathbb{R}.$$ Mean $E[X] = \mu$, variance $\operatorname{Var}(X) = \sigma^{2}$.

19. State the pdf, mean, and variance of the Uniform distribution on $[a,b]$.

    $$f(x) = \frac{1}{b-a}, \quad a \leq x \leq b.$$ Mean $E[X] = \frac{a+b}{2}$, variance $\operatorname{Var}(X) = \frac{(b-a)^{2}}{12}$.

20. Define the joint CDF and joint pdf of a bivariate random vector $(X,Y)$.

    Joint CDF: $F(x,y) = P(X \leq x, Y \leq y)$. For the continuous case the joint pdf $f(x,y)$ satisfies $F(x,y) = \int_{-\infty}^{x}\int_{-\infty}^{y} f(u,v)\,dv\,du$ and $f(x,y) = \frac{\partial^{2} F}{\partial x\,\partial y}$.

21. How do you obtain the marginal distributions from a joint distribution of $(X,Y)$?

    Discrete: $p_X(x) = \sum_{y} p(x,y)$ and $p_Y(y) = \sum_{x} p(x,y)$. Continuous: $f_X(x) = \int_{-\infty}^{\infty} f(x,y)\,dy$ and $f_Y(y) = \int_{-\infty}^{\infty} f(x,y)\,dx$.

22. Define the conditional density of $Y$ given $X=x$.

    For $f_X(x) > 0$, $$f_{Y\mid X}(y \mid x) = \frac{f(x,y)}{f_X(x)}.$$ It is a valid density in $y$ for each fixed $x$, integrating to 1.

23. When are random variables $X$ and $Y$ independent in terms of their joint distribution?

    $X$ and $Y$ are independent iff the joint factorizes: $F(x,y) = F_X(x)F_Y(y)$ for all $x,y$, equivalently $f(x,y) = f_X(x)f_Y(y)$ (or $p(x,y)=p_X(x)p_Y(y)$). Then $f_{Y\mid X}(y\mid x) = f_Y(y)$.

24. Define the expectation of a random variable in the discrete and continuous cases.

    Discrete: $E[X] = \sum_{x} x\,p(x)$. Continuous: $E[X] = \int_{-\infty}^{\infty} x\,f(x)\,dx$, provided the sum/integral converges absolutely. More generally $E[g(X)] = \sum_x g(x)p(x)$ or $\int g(x)f(x)\,dx$ (law of the unconscious statistician).

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