Preface
Why study probability
and simulation
?
Symbulate
Don’t do what Donny Don’t does
About this book
1
What is Probability?
1.1
Instances of randomness
1.2
Interpretations of probability
1.2.1
Relative frequency
1.2.2
Subjective probability
1.3
Working with probabilities
1.3.1
Consistency requirements
1.3.2
Odds
1.4
Approximating probabilities - a brief introduction to simulation
1.5
Why study coins, dice, cards, and spinners?
2
The Language of Probability
2.1
Sample space of outcomes
2.2
Events
2.3
Random variables
2.3.1
Transformations of random variables
2.4
Probability spaces
2.4.1
Probability measures in a dice rolling example
2.4.2
Uniform probability measures #{sec-uniform-prob}
2.4.3
Non-uniform probability measures
2.5
Introduction to simulation
2.5.1
Tactile simulation: Boxes and spinners
2.5.2
Technology simulation: Symbulate
2.5.3
Approximating probabilities - margin of error
2.5.4
Brief summary of Symbulate commands
2.6
Some examples
2.6.1
A weighted die
2.6.2
More dice rolling
2.6.3
Proportion of coin flips immediately following heads that result in heads
2.6.4
Outcomes on a continuous scale
2.6.5
A logarithmic transformation
2.6.6
Continuous analog of rolling two dice
2.6.7
SAT Math scores
2.6.8
SAT Math and Reading scores
2.6.9
One spinner to rule them all?
2.7
Distributions of random variables
2.7.1
Discrete random variables: probability mass functions
2.7.2
Continuous random variables: probability density functions
2.7.3
Cumulative distribution functions
2.7.4
Distributions of transformations
2.7.5
Quantile functions and “universality of the uniform”
2.7.6
Joint distributions
2.7.7
Mixed discrete and continuous random variables
2.8
Expected Values
2.8.1
“Law of the unconscious statistician”
2.8.2
Linearity of expected value
2.9
Variance and covariance
2.9.1
Variance and standard deviation
2.9.2
Covariance and correlation
2.9.3
Variance of sums
2.9.4
Other moments
3
Conditioning
3.1
Conditional probability
3.1.1
Simulating conditional probabilities
3.1.2
Joint, conditional, and marginal probabilities
3.1.3
Multiplication rule
3.1.4
Law of total probability
3.1.5
Bayes rule
3.1.6
Conditional probabilities are probabilities
3.1.7
Conditional versus unconditional probability
3.2
Conditional distributions
3.2.1
Discrete RVs: Conditional pmf
3.2.2
Continuous RVs: Conditional pdf
3.3
Conditional expectation
3.3.1
Law of total expectation
3.3.2
“Taking out what is known”
3.4
Independence
3.4.1
Independence of events
3.4.2
Independence versus uncorrelatedness
3.5
Conditional independence
Appendix
A
Summary of common distributions
References
An Introduction to Probability and Simulation
Chapter 2
The Language of Probability
This chapter introduces the fundamental terminology, objects, and mathematics of random phenomena.