MATH 3033. Introduction to Probability Theory with Lab

Review

Probability of events

experiment and outcome

sample space and event, as sets

probability function, P

conditional probability

independent events

the rules of addition and multiplication

the law of total probability, Bayes' rule

Random variables and probability distributions

random variables

• discrete
• continuous
• independent

functions of random variable

• probability mass function, p
• probability density function, f
• cumulative distribution function, F
• joint and marginal functions

discrete distributions

• Bernoulli distribution, Ber(p)
• Binomial distribution, Bin(n, p)
• Poisson distribution, Pois(μ)
• Geometric distribution, Geo(p)

continuous distributions

• Uniform distribution, U(α, β)
• Exponential distribution, Exp(λ)
• Pareto distribution, Par(α)
• Normal distribution, N(μ, σ²)

distribution features

• expectation, of variables and functions
• variance and standard deviation
• second moment
• covariance, correlation, and correlation coefficient
• median, quantile, and percentile

simulation

• random number generator
• simulating discrete distributions
• simulating continuous distributions

Statistical methods

dataset, random sample, statistical model

graphical summary, as approximate distribution functions

• relative frequency
• histograms
• kernel density functions
• empirical cumulative distribution functions
• scatter plots

sample statistics, as approximate distribution features

• sample mean
• sample variance and sample standard derivation
• sample median and MAD
• empirical quantile, quartile, and the IQR
• five-number summary, box-and-whisker plot
• The Law of Large Numbers

parameter estimation

• estimate and estimator
• unbiasedness, efficiency, and mean squared error
• maximum likelihood and loglikelihood
• least squares estimation and linear regression