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Preface
Why study probability
and simulation
?
Learn by doing
Don’t do what Donny Don’t does
Symbulate
About this book
1
What is Probability?
1.1
Instances of randomness
1.2
Interpretations of probability
1.2.1
Long run relative frequency
1.2.2
Subjective probability
1.3
Working with probabilities
1.3.1
Consistency requirements
1.3.2
Odds
1.4
Proportional reasoning and tables of counts
1.5
Probability of what?
1.6
Approximating probabilities - a brief introduction to simulation
1.7
Why study coins, dice, cards, and spinners?
2
The Language of Probability
2.1
Sample space of outcomes
2.1.1
Summary
2.2
Events
2.2.1
The collection of events of interest
2.2.2
Summary
2.3
Random variables
2.3.1
A random variable is a function
2.3.2
Events involving random variables
2.3.3
Transformations of random variables
2.3.4
Indicator random variables and counting
2.3.5
Summary
2.4
Probability spaces
2.4.1
Some probability measures for a roll of a four-sided
2.4.2
Properties of probability measures
2.4.3
Equally likely outcomes
2.4.4
Uniform probability measures
2.4.5
Non-uniform probability measures
2.5
Probability distributions (a brief introduction)
3
Simulation
3.1
Tactile simulation: Boxes and spinners
3.1.1
Long run averages
3.2
Computer simulation: Symbulate
3.2.1
Simulating outcomes
3.2.2
Simulating random variables
3.2.3
Approximating distributions
3.2.4
Approximating long run averages
3.2.5
Simulating events
3.2.6
Simulating multiple random variables
3.2.7
Simulating outcomes and random variables
3.2.8
Simulating equally likely outcomes
3.2.9
Brief summary of Symbulate commands
3.2.10
Exercises
3.3
Approximating probabilities: Simulation margin of error
3.4
Non-equally likely outcomes: A weighted die
3.5
Simulating from distributions: rolling dice yet again
3.6
Outcomes on a continuous scale: Uniform distributions
3.7
Normal distributions and standard deviation: SAT scores
3.7.1
Standard deviation
3.8
Transformations of random variables
3.8.1
Linear rescaling
3.8.2
Nonlinear transformations
3.8.3
Transformations of multiple random variables
3.9
Joint Normal Distributions: SAT Math and Reading scores
3.10
One spinner to rule them all?
4
Distributions
4.1
Do not confuse a random variable with its distribution
4.2
Discrete random variables: Probability mass functions
4.2.1
Joint probability mass functions
4.3
Continuous random variables: Probability density functions
4.3.1
Joint probability density fuctions
4.4
Cumulative distribution functions
4.4.1
Quantile functions
4.4.2
Universality of the Uniform (One spinner to rule them all)
4.5
Distributions of transformations of random variables
4.5.1
Transformations of multiple random variables
5
Expected Values
5.1
“Expected” value
5.2
“Law of the unconscious statistician” (LOTUS)
5.3
Variance and standard deviation
5.3.1
Standardization
5.4
Probability inequalitlies
5.4.1
Markov’s inequality
5.4.2
Chebyshev’s inequality
5.5
Covariance and correlation
5.6
Expected values of linear combinations of random variables
5.6.1
Linear rescaling
5.6.2
Linearity of expected value
5.6.3
Variance of linear combinations of random variables
5.6.4
Bilinearity of covariance
6
Conditioning
6.1
Conditional probability
6.1.1
Simulating conditional probabilities
6.1.2
Joint, conditional, and marginal probabilities
6.1.3
Multiplication rule
6.1.4
Law of total probability
6.1.5
Bayes rule
6.1.6
Conditioning is “slicing and renormalizing”
6.1.7
Conditional probabilities are probabilities
6.1.8
Conditional versus unconditional probability
6.2
Conditional distributions
6.2.1
Discrete random variables: Conditional probability mass functions
6.2.2
Continuous random variables: Conditional probability density functions
6.3
Independence
6.3.1
Independence of events
6.3.2
Independence of random variables
6.3.3
Independent versus uncorrelated
6.4
Conditional expected value
6.4.1
Conditional expected value as a random variable
6.4.2
Linearity of conditional expected value
6.4.3
Law of total expectation
6.4.4
Taking out what is known
6.4.5
Independent, uncorrelated, and something in between
7
Common Distributions of Discrete Random Variables
7.1
Equally Likely Outcomes and Counting Rules
7.1.1
Summary
7.2
Hypergeometric distributions
7.3
Binomial distributions
7.4
Multinomial distributions
7.5
Negative Binomial distributions
7.6
Poisson distributions
7.6.1
Poisson approximation
References
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An Introduction to Probability and Simulation
7.4
Multinomial distributions