Probability Theory

10 Pages

English

Undergraduate

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Foundations of Probability

1. Experiments, Outcomes, and Sample Spaces

2. Events and Set Operations

3. Probability Axioms and Core Properties

4. Counting with Permutations and Combinations

5. Conditional Probability and Independence

Random Variables and Distributions

6. Discrete Random Variables and Probability Mass Functions

7. Continuous Random Variables and Probability Density Functions

8. Cumulative Distribution Functions

9. Expectation, Variance, and Covariance

10. Common Distributions and the Law of Large Numbers

1. Experiments, Outcomes, and Sample Spaces

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Why is “odd number” not usually an outcome when rolling a die?

Can a sample space have infinitely many outcomes?

Probability experiment:

Outcome:

Sample space:

Elementary outcome:

Probability experiment:

Outcome:

Sample space:

Elementary outcome:

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2. Events and Set Operations

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Does “or” in probability normally exclude the possibility that both events occur?

Can two events be disjoint and independent?

Event:

Union:

Intersection:

Complement:

Event:

Union:

Intersection:

Complement:

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3. Probability Axioms and Core Properties

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Why must every probability be at most 1?

When can probabilities simply be added?

Probability axiom:

Additivity:

Equally likely outcomes:

Inclusion-exclusion rule:

Probability axiom:

Additivity:

Equally likely outcomes:

Inclusion-exclusion rule:

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4. Counting with Permutations and Combinations

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Why does a committee use combinations rather than permutations?

What does \(0!=1\) accomplish?

Multiplication principle:

Factorial:

Permutation:

Combination:

Multiplication principle:

Factorial:

Permutation:

Combination:

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5. Conditional Probability and Independence

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Why are draws without replacement usually dependent?

Can the expression \(P(A\mid B)\) be used when \(P(B)=0\)?

Conditional probability:

Multiplication rule:

Independent events:

Dependent events:

Conditional probability:

Multiplication rule:

Independent events:

Dependent events:

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6. Discrete Random Variables and Probability Mass Functions

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Can a discrete random variable have infinitely many possible values?

Why are probabilities added for \(P(X\geq 2)\)?

Discrete random variable:

Probability mass function:

Support:

Normalization:

Discrete random variable:

Probability mass function:

Support:

Normalization:

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7. Continuous Random Variables and Probability Density Functions

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Can a probability density be greater than one?

Why does \(P(X=c)=0\) not make observing \(c\) impossible?

Continuous random variable:

Probability density function:

Uniform distribution:

Area interpretation:

Continuous random variable:

Probability density function:

Uniform distribution:

Area interpretation:

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8. Cumulative Distribution Functions

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How can a cdf be used when no density exists?

Why is \(P(a<X\leq b)\) expressed with two cdf values?

Cumulative distribution function:

Nondecreasing:

Jump size:

Right-continuity:

Cumulative distribution function:

Nondecreasing:

Jump size:

Right-continuity:

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9. Expectation, Variance, and Covariance

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Why are deviations squared in the variance?

Does covariance equal zero mean the variables are independent?

Expectation:

Variance:

Standard deviation:

Covariance:

Expectation:

Variance:

Standard deviation:

Covariance:

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10. Common Distributions and the Law of Large Numbers

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Does the law of large numbers say that outcomes must alternate to balance previous results?

When is a binomial model appropriate?

Bernoulli distribution:

Binomial distribution:

Poisson distribution:

Law of large numbers:

Bernoulli distribution:

Binomial distribution:

Poisson distribution:

Law of large numbers:

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