Statistics

## Probability

Axiomatic definition of probability and properties

Conditional probability

Multiplication rule

Theorem of total probability

Bayes' theorem and independence of events

## Random Variables

Probability mass function

Probability density function and cumulative distribution functions

Distribution of a function of a random variable

Mathematical expectation

Moments and moment generating function

Chebyshev's inequality

## Standard Distributions

Binomial

Negative binomial

Geometric

Poisson

Hypergeometric

Hniform

Exponential

Gamma, beta and normal distributions

Poisson and normal approximations of a binomial distribution

## Limit Theorems

Weak law of large numbers

Central limit theorem (i.i.d.with finite variance case only)

## Joint Distributions

Joint, marginal and conditional distributions

Distribution of functions of random variables

Joint moment generating function

Product moments, correlation, simple linear regression

Independence of random variables

## Sampling distributions

Chi-square

t and F distributions, and their properties.

## Estimation

Unbiasedness, consistency and efficiency of estimators

method of moments and method of maximum likelihood

Sufficiency, factorization theorem

Completeness, Rao-Blackwell and Lehmann-Scheffe theorems

uniformly minimum variance unbiased estimators

Rao-Cramer inequality

Confidence intervals for the parameters of univariate normal, two independent normal, and one parameter exponential distributions

## Testing of Hypotheses

Basic concepts

applications of Neyman-Pearson Lemma for testing simple and composite hypotheses

Likelihood ratio tests for parameters of univariate normal distribution.