Probability and Statistics I
Year 1 · 7 sections · 77 concepts · 212 practice questions
What's covered
free1 · Counting and probability10 concepts
- Overview 1 · The plan of this section
- Definition 1.1 · Sample spaces and events
- Definition 1.2 · The axioms of probability
- Proposition 1.3 · Consequences of the axioms
- Proposition 1.4 · Equally likely outcomes
- Proposition 1.5 · The multiplication principle and permutations
- Proposition 1.6 · Combinations
- Technique 1.7 · Counting for probability
- Theorem 1.8 · Inclusion–exclusion for probabilities
- Check 1.9 · Check: counting problems
2 · Conditional probability12 concepts
- Overview 2 · The plan of this section
- Definition 2.1 · Conditional probability
- Proposition 2.2 · Conditioning gives a probability
- Proposition 2.3 · The multiplication rule
- Theorem 2.4 · The law of total probability
- Theorem 2.5 · Bayes' theorem
- Example 2.6 · Medical tests and the base-rate fallacy
- Definition 2.7 · Independence
- Proposition 2.8 · Complements of independent events
- Example 2.9 · The Monty Hall problem
- Example 2.10 · Pairwise independent but not independent
- Check 2.11 · Check: conditional probability problems
3 · Discrete random variables13 concepts
- Overview 3 · The plan of this section
- Definition 3.1 · Random variables and probability mass functions
- Definition 3.2 · The distribution function
- Definition 3.3 · Expectation
- Theorem 3.4 · Linearity of expectation
- Proposition 3.5 · Variance
- Proposition 3.6 · Bernoulli and binomial distributions
- Proposition 3.7 · The geometric distribution
- Proposition 3.8 · The Poisson distribution
- Theorem 3.9 · Poisson approximation to the binomial
- Example 3.10 · The indicator method
- Example 3.11 · The hypergeometric distribution
- Check 3.12 · Check: choosing a discrete distribution
4 · Continuous random variables10 concepts
- Overview 4 · The plan of this section
- Definition 4.1 · Probability density functions
- Proposition 4.2 · Density and distribution function
- Definition 4.3 · Expectation and variance of continuous variables
- Proposition 4.4 · The uniform distribution
- Proposition 4.5 · The exponential distribution
- Proposition 4.6 · The normal distribution
- Proposition 4.7 · Standardising a normal variable
- Technique 4.8 · Transformations of continuous random variables
- Check 4.9 · Check: continuous distributions
5 · Joint distributions11 concepts
- Overview 5 · The plan of this section
- Definition 5.1 · Joint distributions and marginals
- Definition 5.2 · Independent random variables
- Proposition 5.3 · Expectations of products of independent variables
- Definition 5.4 · Covariance and correlation
- Proposition 5.5 · Variance of a sum
- Proposition 5.6 · Correlation lies in [−1, 1]
- Example 5.7 · Uncorrelated but dependent
- Theorem 5.8 · Sums of independent random variables
- Definition 5.9 · Conditional distributions and expectation
- Check 5.10 · Check: working with several random variables
6 · Limit theorems9 concepts
- Overview 6 · The plan of this section
- Theorem 6.1 · Markov's inequality
- Theorem 6.2 · Chebyshev's inequality
- Theorem 6.3 · The weak law of large numbers
- Theorem 6.4 · The central limit theorem
- Technique 6.5 · Normal approximation to the binomial
- Example 6.6 · The sampling distribution of the mean
- Example 6.7 · Polls and the margin of error
- Check 6.8 · Check: using the limit theorems
7 · Statistics12 concepts
- Overview 7 · The plan of this section
- Definition 7.1 · Samples and statistics
- Definition 7.2 · Estimators, bias and mean squared error
- Proposition 7.3 · The sample variance is unbiased
- Definition 7.4 · Likelihood and maximum likelihood
- Example 7.5 · Standard maximum likelihood estimates
- Definition 7.6 · Confidence intervals
- Proposition 7.7 · Unknown variance and the t distribution
- Definition 7.8 · Hypothesis tests and p-values
- Example 7.9 · The z-test for a mean
- Remark 7.10 · What a p-value is not
- Check 7.11 · Check: estimation and testing
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Proposition 1.3 · Consequences of the axioms
Complement: ("at least one" = "none"). Union: add, subtract the overlap. Monotone: bigger events are more likely.
Proof
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