Chapter 2
Random Variables and Probability Distributions
In many experiments, we are not interested in the specific outcome itself, but rather in a numerical value associated with that outcome. For example, if we test three electronic components, we may not care which specific ones fail, but only the total number of failures
2.1 Random Variables
2.1.1 Summation methods you will need
2.1.2 Practice problems
2.2 Discrete Random Variables
2.2.1 Practice problems
2.2.2 Practice problems
2.3 Continuous Random Variables
2.3.1 Practice problems
2.4 Mathematical Expectations and Generating Functions of Random Variables
2.4.1 Expectations and Moments
2.4.2 Probability Generating Functions (PGFs)
2.4.3 Moment Generating Functions (MGFs)
2.4.4 Characteristic Functions
2.4.5 Practice problems
2.5 Some Important Discrete Probability Mass Function
2.5.1 The Bernoulli Distribution
2.5.2 Discrete Uniform
2.5.3 The Binomial Distribution
2.5.4 The Hypergeometric Distribution
2.5.5 The Geometric Distribution
2.5.6 The Negative Binomial Distribution
2.5.7 The Poisson Distribution
2.5.8 The Zeta (or Zipf) Distribution
2.5.9 Practice problems
2.6 Some Important Continuous Probability Density Functions
2.6.1 The Uniform Distribution
2.6.2 Exponential Distribution
2.6.3 Gamma Distribution
2.6.4 Beta Distribution
2.6.5 Normal Distribution
2.6.6 Cauchy Distribution
2.6.7 The Weibull Distribution
2.6.8 Rayleigh Distribution
2.6.9 The Pareto Distribution
2.6.10 Practice problems
2.7 Cumulative Distribution Function (CDF)
2.7.1 Practice problems
2.8 Markov’s and Chebyshev’s Inequality
2.8.1 Practice problems
2.1.1 Summation methods you will need
2.1.2 Practice problems
2.2 Discrete Random Variables
2.2.1 Practice problems
2.2.2 Practice problems
2.3 Continuous Random Variables
2.3.1 Practice problems
2.4 Mathematical Expectations and Generating Functions of Random Variables
2.4.1 Expectations and Moments
2.4.2 Probability Generating Functions (PGFs)
2.4.3 Moment Generating Functions (MGFs)
2.4.4 Characteristic Functions
2.4.5 Practice problems
2.5 Some Important Discrete Probability Mass Function
2.5.1 The Bernoulli Distribution
2.5.2 Discrete Uniform
2.5.3 The Binomial Distribution
2.5.4 The Hypergeometric Distribution
2.5.5 The Geometric Distribution
2.5.6 The Negative Binomial Distribution
2.5.7 The Poisson Distribution
2.5.8 The Zeta (or Zipf) Distribution
2.5.9 Practice problems
2.6 Some Important Continuous Probability Density Functions
2.6.1 The Uniform Distribution
2.6.2 Exponential Distribution
2.6.3 Gamma Distribution
2.6.4 Beta Distribution
2.6.5 Normal Distribution
2.6.6 Cauchy Distribution
2.6.7 The Weibull Distribution
2.6.8 Rayleigh Distribution
2.6.9 The Pareto Distribution
2.6.10 Practice problems
2.7 Cumulative Distribution Function (CDF)
2.7.1 Practice problems
2.8 Markov’s and Chebyshev’s Inequality
2.8.1 Practice problems