Course Overview
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Course Synopsis
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Statistical methods used in practice are based on a foundation of statistical theory. One branch of this theory uses the tools of probability to establish important distributional results that are used throughout statistics. This course will cover the mathematical foundation for the Probability theory, its uses and applications.
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Course Learning Outcomes
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At the end of the course, you should be able to:
- Understand of the principles of probability theory
- Basic probability axioms and rules and the moments of discrete and continuous random variables
- Recognize common probability distributions for discrete and continuous variables
- Apply methods from algebra and calculus to derive the mean, variance and other measures for a range of probability distributions
- How to calculate probabilities, and derive the marginal and conditional distributions of bivariate random variables
- Calculate probabilities relevant to multivariate distributions, including marginal and conditional probabilities and the covariance of two random variables
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Course Calendar
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Week 01
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1
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Probability - Some Basic Terms
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2
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Random Variable - Concept
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3
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Random Variable - Example
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4
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Random Variable - Discrete Data
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5
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Probability Mass Function (PMF) - Concept
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6
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Probability Mass Function (PMF) - Example
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7
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Random Variable - Continuous Data
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8
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Probability Density Function (PDF)
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9
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Probability Generating function (PGF)
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10
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Mean and Variance of PGF
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12
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Linear Combination of PDFs
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13
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Cumulative Distribution Function (CDF) - Concept
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14
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Cumulative Distribution Function (CDF) - Example Discrete Case
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15
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Obtaining PMF from CDF
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16
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Cumulative Distribution Function (CDF) - Example Continoius Case
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17
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Two Random Variables Being Equal in Distribution
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Week 02
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18
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Cumulative Distribution Function (CDF) - First Property
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19
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Cumulative Distribution Function (CDF) - Second Property
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20
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Cumulative Distribution Function (CDF) - Third Property
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21
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Cumulative Distribution Function (CDF) - Fourth Property
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22
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Evaluating Probabilities using CDF
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23
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Derivation of CDF is PDF
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24
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In Continuous case, Probability does not exist at ANY particular point
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25
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Monotonicity - Concept
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26
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Total Probability is one
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27
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Finding an Unknown constant from PMF
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28
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Finding an Unknown constant from PDF
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29
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Transformation of Discrete variables - one to one case
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30
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Transformation of Discrete variables - NOT one to one case
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31
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Transformation of Continuous variables
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32
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Transformation of Continuous variables - Jacobian Transformation
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33
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Transformation of Continuous variables - Jacobian Transformation - Example 1
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34
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Transformation of Continuous variables - Jacobian Transformation - Example 2
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Quiz-1
Week 03
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35
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Mode of Discrete variable
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36
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Mode of Continuous variable
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37
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Median of Discrete variable
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38
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Median of Continuous variable
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39
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Percentile of Continuous variable - Concept
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40
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Percentile of Continuous variable - Example
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41
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Inverse CDF (Quantile Function)
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42
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One random variable is larger than other random variable
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43
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Mathematical Expectation of variable - Concept
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44
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Mathematical Expectation of a function - Concept
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45
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Mathematical Expectation - Linear Combination of Expected Values
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46
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Mathematical Expectation of a function - Discrete case example 1
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47
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Mathematical Expectation of a function - Discrete case example 2
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48
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Mean, Variance and Standard Deviation of random variable - A theorem
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49
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Mean, Variance and Standard Deviation of random variable - Discrete case
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50
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Degenerate random variable - Concept
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51
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Mean of the symmetric distribution - Proof
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52
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Mean and variance of standardized variable - Proof
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Week 04
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55
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Moment Generating Function (MGF) - Concept
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56
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Moment Generating Function (MGF) of some well-known distributions
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57
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General Rule for the product of Expected Value
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58
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Mathematical Expectation - Another way of computing
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59
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Derivation of Mean and Variance from MGF
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60
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Derivation of m-th Moment from MGF
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61
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Derivation of m-th Moment about an Arbitrary origin from MGF
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62
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Alternative Method of finding moments from MGF Using Maclaurins's Series
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63
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Relationship between MGF of Standardized variable and MGF of original variable - Theorem 1
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64
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Relationship between MGF of Standardized variable and MGF of original variable - Theorem 2
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65
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Cumulant Generating Function (CGF)
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66
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Cumulant Generating Function (CGF) - Additive Property
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67
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Relationship between Central MGF and CGF
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Assignment-1
Week 05
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68
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Chevbyshev's Inequality - Concept
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69
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Chevbyshev's Inequality - Example
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70
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Chevbyshev's Inequality - Another form of presenting it
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71
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Relationship between Harmonic Mean, Geometric Mean and Arithmetic Mean
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72
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Random Vector - Concept
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73
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Event in case of a two-dimensional space - Concept
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74
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Joint Cumulative Distribution Function
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75
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Discrete Random Vector and Joint Probability Mass Function
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76
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Support of Discrete Random Vector - Concept
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77
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Continuous Random Vector and Joint Probability Density Function
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78
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Determination of the probability of an event - Continuous Vector
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79
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Support of Continuous Random Vector - Concept
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80
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Properties of joint cumulative probability distribution function (Part 1)
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81
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Properties of joint cumulative probability distribution function (Part 2)
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Week 06
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82
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Marginal Probability Mass Function - Concept
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83
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Marginal Probability Mass Function - Example
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84
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Marginal Probability Density Function - Concept
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85
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Computations of Probabilities that cannot be found through Marginal PDFs
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86
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Expected Value of a function of a Random Vector
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87
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Computations of Probabilities for Continuous Random Vector
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88
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Expectation of the product of the two random vectors - Discrete case
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89
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Expectation of the product of the two random vectors - Continuous case
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90
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Expectation of the Ratio of two random vectors - Continuous case
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91
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Moments from MGF of Random Vector
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92
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Expected value of Random Vector
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93
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Linear Combination of Expected Values of Function of a Random Vector
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Quiz-2
Week 07
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94
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Transformation of a Bivariate Probability Mass Function - Example 1
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95
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Transformation of a Bivariate Probability Mass Function - Example 2
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96
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Transformation of a Bivariate Probability Density Function - Using Jacobean
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97
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Transformation of a Bivariate Probability Mass Function - Using MGF
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98
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Characteristic Function
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99
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Derivation of Mean and Variance of a distribution using Characteristic Function
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100
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Conditional Distribution - Concept
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101
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Conditional Expectation Conditional Mean and Variance
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102
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Conditional Expectation - Theorem
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Week 08
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103
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Correlation Coefficient of Bivariate Distribution
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104
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Correlation Coefficient - Properties (1st)
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105
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Correlation Coefficient - Properties (2nd, 3rd)
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106
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Correlation Coefficient - Properties (4th, 5th, 6th)
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107
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Correlation Coefficient - Computation in case of joint PDF
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Mid Term Exams
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108
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Conditional Distribution - Variance of joint PDF of two random variables - Example
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109
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Conditional mean of Y given X that is LINEAR in X
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110
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Independent Random Variables - Concept
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111
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Independent Random Variables - Example
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112
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Independent Random Variables - Theorem X and Y are independent if and only if F(x,y) = FX(x)FY(y)
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113
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Independent Random Variables - Theorem X and Y are independent then E[u(X1)v(X2)] = E[u(X1)] E[v(X2)]
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114
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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)
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Week 09
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115
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Discrete Uniform Distribution - Concept and Example
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116
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Discrete Uniform Distribution - Probability Mass Function PMF
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117
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Discrete Uniform Distribution - Cumulative Distribution Function CDF
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118
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Discrete Uniform Distribution - Derivation of the Mean and Variance
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119
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Discrete Uniform Distribution - Derivation of the MGF
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120
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Binomial Distribution - Binomial experiment and PMF
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121
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Binomial Distribution – Example
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122
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Binomial Distribution - Shape of the distribution
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123
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Binomial Distribution - Another Example
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124
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Binomial Distribution - Mean and Variance with example
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125
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Binomial Distribution - Derivation of the MGF
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126
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Binomial Distribution - Recognizing the parameters by its MGF
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127
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Binomial Distribution - The sum of m independent Binomial random variables with identical p is also Binomial
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Week 10
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128
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Negative Binomial Distribution PMF and shape of the distribution
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129
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Negative Binomial Distribution - Application and Example
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130
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Geometric Distribution - PMF and shape of the distribution
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131
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Geometric Distribution - Application and example
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132
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Geometric Distribution - Mean and Variance with example
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133
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Multinomial Distribution
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134
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Multinomial Distribution - Binomial as a Special Case
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135
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Multinomial distribution - Application and example
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136
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Multinomial Distribution - MGF of the Trinomial Distribution as a special case of the Multinomial Distribution
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Week 11
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137
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Hypergeometric Distribution - PMF and shape of the distribution
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138
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Hypergeometric Distribution – Example
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139
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Hypergeometric Distribution - Derivation of the Mean
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140
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Hypergeometric Distribution - Derivation of the Variance
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Assignment-2
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141
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Poisson Distribution - PMF and shape of the distribution
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142
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Poisson Distribution - Poisson Process
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143
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Poisson Distribution - Poisson Process - Application and example
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144
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Poisson Distribution - Mean, Variance and Coefficient of Variation with example
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Week 12
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145
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Poisson Distribution - Derivation of Mean
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146
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Poisson Distribution - Derivation of Variance
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147
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Poisson Distribution – MGF
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148
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Poisson Distribution - Poisson Approximation to the Binomial distribution - Derivation
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149
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Poisson Distribution - Poisson Approximation to the Binomial distribution - Example
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150
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Continuous Uniform Distribution / Rectangular Distribution - PDF, CDF and shape of the distribution
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151
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Continuous Uniform Distribution - Derivation of Mean and Variance
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152
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Continuous Uniform Distribution - Application and example
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153
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Exponential Distribution - PDF, CDF and shape of the distribution
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154
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Exponential Distribution - Derivation of mean
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155
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Exponential Distribution - Derivation of variance
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156
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Exponential Distribution - Application and example
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157
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Exponential Distribution - Moment Generating Function MGF
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Week 13
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158
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Gamma Distribution - PDF, CDF and shape of the distribution
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159
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Gamma Distribution - Derivation of Mean and Variance
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160
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Gamma Distribution - computation of probabilities with an Example
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161
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Gamma Distribution - Moment Generating Function MGF
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162
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Beta Distribution - PDF, CDF and shape of the distribution
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163
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Beta Distribution - Derivation of Mean and Variance
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164
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Beta distribution - Application and example
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165
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Normal Distribution - PDF, CDF and shape of the distribution
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166
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Normal Distribution - Standard normal random variable - Concept and its PDF
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167
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Normal Distribution - Process of Standardization, Area Table of the Standard Normal Distribution
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168
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Normal Distribution - Computation of some specific percentiles
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Quiz-3
Week 14
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169
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Normal Distribution - Function f(x) of normal distribution is a proper PDF
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170
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Normal Distribution - Derivation of Mean and Variance
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171
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Normal distribution - Relationship between Mean, Median and Mode
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172
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Normal Distribution - Relationship between Quartile Deviation and Standard Deviation
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173
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Normal Distribution - Points of Inflection of Normal curve
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174
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Normal Distribution - Odd order moments about the mean
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175
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Normal Distribution - Even order moments about the mean
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176
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Normal Distribution - Obtaining the first two Raw Moments from its MGF
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177
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Normal Distribution - Cumulant Generating Function and its utilization for finding its cumulants
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178
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Normal Distribution - Normal Approximation to the Binomial distribution
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179
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Normal Distribution - Normal Approximation to the Poisson distribution
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Week 15
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180
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Bivariate Normal Distribution - PDF and shape of the distribution
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181
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Bivariate Normal Distribution - Detailed Discussion of the Shape of the Distribution
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182
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Bivariate Normal Distribution - Computation of probabilities with an example
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183
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Bivariate Normal Distribution - Moment Generating Function MGF
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184
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Bivariate Normal Distribution - Marginal Distributions are themselves Normal
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185
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Bivariate Normal Distribution - X and Y are independent IF AND ONLY IF the Correlation Coefficient ? = 0
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186
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Bivariate Normal Distribution - Conditional Distributions are themselves Normal
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187
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Bivariate Normal Distribution - Conditional Distributions are themselves Normal - Graphical Interpretation
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Final Term exam
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