# Virtual University of Pakistan

## STA642 : Probability Distributions

 Your like has been submitted! Error! Please try again. I like this Course

### Course Info

#### Course Category

Probability & Statistics

3

N/A

#### Instructor

Course Recorded by: Dr. Saleha Naghmi Habibullah
PhD

### Course Contents

Probability - Some Basic Terms Random Variable - Concept Random Variable - Example Random Variable - Discrete Data Probability Mass Function (PMF) - Concept Probability Mass Function (PMF) - Example Random Variable - Continuous Data Probability Density Function (PDF) Probability Generating function (PGF) Mean and Variance of PGF Some well-known PGF Linear Combination of PDFs Cumulative Distribution Function (CDF) - Concept Cumulative Distribution Function (CDF) - Example Discrete Case Obtaining PMF from CDF Cumulative Distribution Function (CDF) - Example Continoius Case Two Random Variables Being Equal in Distribution Cumulative Distribution Function (CDF) - First Property Cumulative Distribution Function (CDF) - Second Property Cumulative Distribution Function (CDF) - Third Property Cumulative Distribution Function (CDF) - Fourth Property Evaluating Probabilities using CDF Derivation of CDF is PDF In Continuous case, Probability does not exist at ANY particular point Monotonicity - Concept Total Probability is one Finding an Unknown constant from PMF Finding an Unknown constant from PDF Transformation of Discrete variables - one to one case Transformation of Discrete variables - NOT one to one case Transformation of Continuous variables Transformation of Continuous variables - Jacobian Transformation Transformation of Continuous variables - Jacobian Transformation - Example 1 Transformation of Continuous variables - Jacobian Transformation - Example 2 Mode of Discrete variable Mode of Continuous variable Median of Discrete variable Median of Continuous variable Percentile of Continuous variable - Concept Percentile of Continuous variable - Example Inverse CDF (Quantile Function) One random variable is larger than other random variable Mathematical Expectation of variable - Concept Mathematical Expectation of a function - Concept Mathematical Expectation - Linear Combination of Expected Values Mathematical Expectation of a function - Discrete case example 1 Mathematical Expectation of a function - Discrete case example 2 Mean, Variance and Standard Deviation of random variable - A theorem Mean, Variance and Standard Deviation of random variable - Discrete case Degenerate random variable - Concept Mean of the symmetric distribution - Proof Mean and variance of standardized variable - Proof Moments - concept Moments Ratios Moment Generating Function (MGF) - Concept Moment Generating Function (MGF) of some well-known distributions General Rule for the product of Expected Value Mathematical Expectation - Another way of computing Derivation of Mean and Variance from MGF Derivation of m-th Moment from MGF Derivation of m-th Moment about an Arbitrary origin from MGF Alternative Method of finding moments from MGF Using Maclaurins's Series Relationship between MGF of Standardized variable and MGF of original variable - Theorem 1 Relationship between MGF of Standardized variable and MGF of original variable - Theorem 2 Cumulant Generating Function (CGF) Cumulant Generating Function (CGF) - Additive Property Relationship between Central MGF and CGF Chevbyshev's Inequality - Concept Chevbyshev's Inequality - Example Chevbyshev's Inequality - Another form of presenting it Relationship between Harmonic Mean, Geometric Mean and Arithmetic Mean Random Vector - Concept Event in case of a two-dimensional space - Concept Joint Cumulative Distribution Function Discrete Random Vector and Joint Probability Mass Function Support of Discrete Random Vector - Concept Continuous Random Vector and Joint Probability Density Function Determination of the probability of an event - Continuous Vector Support of Continuous Random Vector - Concept Properties of joint cumulative probability distribution function (Part 1) Properties of joint cumulative probability distribution function (Part 2) Marginal Probability Mass Function - Concept Marginal Probability Mass Function - Example Marginal Probability Density Function - Concept Computations of Probabilities that cannot be found through Marginal PDFs Expected Value of a function of a Random Vector Computations of Probabilities for Continuous Random Vector Expectation of the product of the two random vectors - Discrete case Expectation of the product of the two random vectors - Continuous case Expectation of the Ratio of two random vectors - Continuous case Moments from MGF of Random Vector Expected value of Random Vector Linear Combination of Expected Values of Function of a Random Vector Transformation of a Bivariate Probability Mass Function - Example 1 Transformation of a Bivariate Probability Mass Function - Example 2 Transformation of a Bivariate Probability Density Function - Using Jacobean Transformation of a Bivariate Probability Mass Function - Using MGF Characteristic Function Derivation of Mean and Variance of a distribution using Characteristic Function Conditional Distribution - Concept Conditional Expectation Conditional Mean and Variance Conditional Expectation - Theorem Correlation Coefficient of Bivariate Distribution Correlation Coefficient - Properties (1st) Correlation Coefficient - Properties (2nd, 3rd) Correlation Coefficient - Properties (4th, 5th, 6th) Correlation Coefficient - Computation in case of joint PDF Conditional Distribution - Variance of joint PDF of two random variables - Example Conditional mean of Y given X that is LINEAR in X Independent Random Variables - Concept Independent Random Variables - Example Independent Random Variables - Theorem X and Y are independent if and only if F(x,y) = FX(x)FY(y) Independent Random Variables - Theorem X and Y are independent then E[u(X1)v(X2)] = E[u(X1)] E[v(X2)] Independent Random Variables - Theorem X and Y are independent if and only if the joint MGF show M(t1,t2)= M(t1,0) M(0,t2) Discrete Uniform Distribution - Concept and Example Discrete Uniform Distribution - Probability Mass Function PMF Discrete Uniform Distribution - Cumulative Distribution Function CDF Discrete Uniform Distribution - Derivation of the Mean and Variance Discrete Uniform Distribution - Derivation of the MGF Binomial Distribution - Binomial experiment and PMF Binomial Distribution – Example Binomial Distribution - Shape of the distribution Binomial Distribution - Another Example Binomial Distribution - Mean and Variance with example Binomial Distribution - Derivation of the MGF Binomial Distribution - Recognizing the parameters by its MGF Binomial Distribution - The sum of m independent Binomial random variables with identical p is also Binomial Negative Binomial Distribution PMF and shape of the distribution Negative Binomial Distribution - Application and Example Geometric Distribution - PMF and shape of the distribution Geometric Distribution - Application and example Geometric Distribution - Mean and Variance with example Multinomial Distribution Multinomial Distribution - Binomial as a Special Case Multinomial distribution - Application and example Multinomial Distribution - MGF of the Trinomial Distribution as a special case of the Multinomial Distribution Hypergeometric Distribution - PMF and shape of the distribution Hypergeometric Distribution – Example Hypergeometric Distribution - Derivation of the Mean Hypergeometric Distribution - Derivation of the Variance Poisson Distribution - PMF and shape of the distribution Poisson Distribution - Poisson Process Poisson Distribution - Poisson Process - Application and example Poisson Distribution - Mean, Variance and Coefficient of Variation with example Poisson Distribution - Derivation of Mean Poisson Distribution - Derivation of Variance Poisson Distribution – MGF Poisson Distribution - Poisson Approximation to the Binomial distribution - Derivation Poisson Distribution - Poisson Approximation to the Binomial distribution - Example Continuous Uniform Distribution / Rectangular Distribution - PDF, CDF and shape of the distribution Continuous Uniform Distribution - Derivation of Mean and Variance Continuous Uniform Distribution - Application and example Exponential Distribution - PDF, CDF and shape of the distribution Exponential Distribution - Derivation of mean Exponential Distribution - Derivation of variance Exponential Distribution - Application and example Exponential Distribution - Moment Generating Function MGF Gamma Distribution - PDF, CDF and shape of the distribution Gamma Distribution - Derivation of Mean and Variance Gamma Distribution - computation of probabilities with an Example Gamma Distribution - Moment Generating Function MGF Beta Distribution - PDF, CDF and shape of the distribution Beta Distribution - Derivation of Mean and Variance Beta distribution - Application and example Normal Distribution - PDF, CDF and shape of the distribution Normal Distribution - Standard normal random variable - Concept and its PDF Normal Distribution - Process of Standardization, Area Table of the Standard Normal Distribution Normal Distribution - Computation of some specific percentiles Normal Distribution - Function f(x) of normal distribution is a proper PDF Normal Distribution - Derivation of Mean and Variance Normal distribution - Relationship between Mean, Median and Mode Normal Distribution - Relationship between Quartile Deviation and Standard Deviation Normal Distribution - Points of Inflection of Normal curve Normal Distribution - Odd order moments about the mean Normal Distribution - Even order moments about the mean Normal Distribution - Obtaining the first two Raw Moments from its MGF Normal Distribution - Cumulant Generating Function and its utilization for finding its cumulants Normal Distribution - Normal Approximation to the Binomial distribution Normal Distribution - Normal Approximation to the Poisson distribution Bivariate Normal Distribution - PDF and shape of the distribution Bivariate Normal Distribution - Detailed Discussion of the Shape of the Distribution Bivariate Normal Distribution - Computation of probabilities with an example Bivariate Normal Distribution - Moment Generating Function MGF Bivariate Normal Distribution - Marginal Distributions are themselves Normal Bivariate Normal Distribution - X and Y are independent IF AND ONLY IF the Correlation Coefficient ? = 0 Bivariate Normal Distribution - Conditional Distributions are themselves Normal Bivariate Normal Distribution - Conditional Distributions are themselves Normal - Graphical Interpretation