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117Definition of Discrete Random Variables
118Probability Mass Functions (PMF)
119Properties of Valid PMFs
120Cumulative Distribution Functions (CDF) for Discrete Variables
121Expected Value of Discrete Random Variables
122Variance and Standard Deviation of Discrete Random Variables
123Bernoulli Trial Definition and Properties
124Bernoulli Distribution PMF and Parameters
125Bernoulli Mean and Variance
126From Bernoulli to Binomial: Multiple Trials
127Binomial Distribution PMF
128Binomial Parameters n and p
129Binomial Mean and Variance
130Calculating Binomial Probabilities
131Real-World Applications of Binomial Distributions
132The Geometric Distribution: Waiting for the First Success
133Expectation and Variance of the Geometric Distribution
134Memoryless Property of the Geometric Distribution
135The Negative Binomial Distribution: Waiting for r Successes
136Expectation and Variance of the Negative Binomial
137Geometric vs Negative Binomial: Key Differences
138Real-World Applications: Quality Control and Surveys
139The Poisson Process and Rate Parameter
140Poisson Probability Mass Function
141Mean and Variance of Poisson Distribution
142Poisson as Limit of Binomial
143Cumulative Poisson Probabilities
144Poisson Applications: Arrivals and Events
145Sum of Independent Poisson Variables
146When to Use Poisson vs Other Distributions
147Expected Value of Discrete Random Variables
148Variance and Standard Deviation of Discrete Distributions
149Properties of Expectation and Variance
150Moment Generating Functions
151Expected Value and Variance for Common Distributions
152Decision Making Under Uncertainty
153Real-World Use Cases: Quality Control and Defects
154Real-World Use Cases: Customer Behavior and Events

117Definition of Discrete Random Variables
118Probability Mass Functions (PMF)
119Properties of Valid PMFs
120Cumulative Distribution Functions (CDF) for Discrete Variables
121Expected Value of Discrete Random Variables
122Variance and Standard Deviation of Discrete Random Variables
123Bernoulli Trial Definition and Properties
124Bernoulli Distribution PMF and Parameters
125Bernoulli Mean and Variance
126From Bernoulli to Binomial: Multiple Trials
127Binomial Distribution PMF
128Binomial Parameters n and p
129Binomial Mean and Variance
130Calculating Binomial Probabilities
131Real-World Applications of Binomial Distributions
132The Geometric Distribution: Waiting for the First Success
133Expectation and Variance of the Geometric Distribution
134Memoryless Property of the Geometric Distribution
135The Negative Binomial Distribution: Waiting for r Successes
136Expectation and Variance of the Negative Binomial
137Geometric vs Negative Binomial: Key Differences
138Real-World Applications: Quality Control and Surveys
139The Poisson Process and Rate Parameter
140Poisson Probability Mass Function
141Mean and Variance of Poisson Distribution
142Poisson as Limit of Binomial
143Cumulative Poisson Probabilities
144Poisson Applications: Arrivals and Events
145Sum of Independent Poisson Variables
146When to Use Poisson vs Other Distributions
147Expected Value of Discrete Random Variables
148Variance and Standard Deviation of Discrete Distributions
149Properties of Expectation and Variance
150Moment Generating Functions
151Expected Value and Variance for Common Distributions
152Decision Making Under Uncertainty
153Real-World Use Cases: Quality Control and Defects
154Real-World Use Cases: Customer Behavior and Events

Lesson 149 of 2,145·4. Discrete Probability DistributionsPro lesson

Properties of Expectation and Variance

Master linearity of expectation, variance of linear transformations, and key properties for calculating combined distributions.

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