Experiments/Models, Ideas of deterministic and non-deterministic models. Random Experiment, concept of statistical regularity.
Definitions of: (i) Sample space, (ii) Discrete sample space: finite and countably infinite, (iii) Event, (iv) Elementary event, (v) Complement of an event. (vi) Certain event (vii) Impossible event
Concept of occurrence of an event.
Algebra of events and its representation in set theory notation. Occurrence of following events.
(i) at least one of the given events,
(ii) none of the given events,
(iii) all of the given events,
(iv) mutually exclusive events,
(v) mutually exhaustive events,
(vi) exactly one event out of the given events.
Classical definition of probability and its limitations.
Probability model, probability of an event, equiprobable and non- equiprobable sample space.
Axiomatic definition of probability. Theorems and results on probability with proofs based on axiomatic definition such as P(AUB) = P(A) + P(B) = P(A n B). Generalization
P(A UBU C), 0 ≤ P(A) ≤ 1, P(A) + P(A) = 1, P(0) = 0, P(A) ≤ P(B) when AB Boole's inequality.
3.Conditional Probability and Independence:
Definition of conditional probability of an event. Results on conditional probability.
Definition of independence of two events P(A B) = P(A) · P(B). Pairwise independence and mutual independence for three events. Multiplication theorem
P(A B) = P(A) P(BIA). Generalization to P(ABC).
Partition of the sample space, prior and posterior probabilities. Proof of Bayes' theorem. Applications of Bayes' theorem in real life. True positive, false positive and sensitivity of test as application of Bayes' theorem.
4.Univariate Probability Distributions (Defined on Discrete Sample Space):
Concept and definition of a discrete random variable. Probability mass function (p.m.f.) and cumulative distribution function (c.d.f.), F(-) of discrete random variable, properties of c.d.f.. Mode and median of a univariate discrete probability distribution.
5.Mathematical Expectation (Univariate Random Variable):
Definition of expectation (Mean) of a random variable, expectation of a function of a random variable, m.g.f. and c.g.f. Properties of m.g.f. and c.g.f.
Definitions of variance, standard deviation (s.d.) and Coefficient of variation (C.V.) of univariate probability distribution, effect of change of origin and scale on mean, variance and s.d.
Definition of raw, central and factorial raw moments of univariate probability Distributions and their interrelations (without proof).
Coefficients of skewness and kurtosis based on moments.
6.Bivariate Discrete Probability Distribution:
Definition of two-dimensional discrete random variable, its joint p.m.f. and its distribution function and their properties.
Concept of identically distributed r.v.s.
Computation of probabilities of events in bivariate probability distribution.
Concepts of marginal and conditional probability distributions.
Independence of two discrete random variables based on joint and marginal p.m.f.s.
7.Mathematical Expectation (Bivariate Random Variable):
Definition of raw and central moments, m.g.f., c.g.f.
Theorems on expectations of sum and product of two jointly distributed random variables.
Conditional expectation.
Definition of conditional mean and conditional variance.
Definition of covariance, coefficient of correlation, independence and uncorrelatedness of two variables.
Variance of linear combination of variables Var (ax + by).