Instructor

Course Recorded by: Dr. Saleha Naghmi Habibullah PhD National College of Business Administration & Economics (NCBAE)

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 wellknown 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 wellknown distributions
General Rule for the product of Expected Value
Mathematical Expectation  Another way of computing
Derivation of Mean and Variance from MGF
Derivation of mth Moment from MGF
Derivation of mth 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 twodimensional 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
