Probability models
Build discrete distributions, choose binomial and normal models, and distinguish transformations from independent combinations.
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Before you begin
Build discrete distributions, choose binomial and normal models, and distinguish transformations from independent combinations.
Learning goals
- Build a finite distribution from equally likely outcome counts
- Calculate an event probability from a finite distribution
- Calculate a finite discrete variable's expectation
- Calculate a finite discrete variable's second moment
- Calculate a finite discrete variable's variance
- Build a transformed finite distribution, combining equal values
- Calculate the expectation of a transformed finite variable
- Calculate the variance of a transformed finite variable
- Explain discrete models, probability constraints and moment interpretation
- Represent random variables, expectations and variances
- Choose and use a binomial probability model
- Calculate an event probability in a specified binomial model
- Calculate a conditional probability in a specified binomial model
- Calculate the expectation of a specified binomial count
- Calculate the variance of a specified binomial count
- Recover the unique binomial parameters from exact moments
- Find the least trial count satisfying a binomial probability threshold
- Choose and use a normal probability model
- Calculate an event probability in a specified normal model
- Calculate a conditional probability in a specified normal model
- Recover a normal boundary from its specified tail probability
- Recover a normal mean from a probability and known variance
- Recover a normal variance from a probability and known mean
- Recover the unique normal mean and variance from two probabilities
- Calculate a linear combination's expectation from specified moments
- Calculate a linear combination's variance with justified dependence conditions
- Calculate a probability for a specified combination of normal variables
Syllabus statements covered
- Include: discrete random variables, probability distributions, expectations and variances; binomial distribution B(n, p) as a discrete probability model and conditions for suitability; binomial mean and variance without proof. Exclude finding a cumulative distribution function of a discrete random variable.
- Include: concept of continuous random variables; concept of normal distribution as a continuous probability model with mean and variance; use of N(μ, σ²); standard normal distribution; finding P(X < x₁) or related probabilities given x₁, μ, σ, and finding a relationship between x₁, μ, σ given a probability; symmetry and properties of the normal curve; problems using E(aX + b) and Var(aX + b), and E(aX + bY) and Var(aX + bY) for independent X and Y. Exclude normal approximation to binomial distribution. The concept of continuous random variables is for teaching and learning only.
- Include: discrete random variables, probability distributions, expectations and variances; binomial distribution B(n, p) as a discrete probability model and conditions for suitability; binomial mean and variance without proof. Exclude finding a cumulative distribution function of a discrete random variable.
- Include: concept of continuous random variables; concept of normal distribution as a continuous probability model with mean and variance; use of N(μ, σ²); standard normal distribution; finding P(X < x₁) or related probabilities given x₁, μ, σ, and finding a relationship between x₁, μ, σ given a probability; symmetry and properties of the normal curve; problems using E(aX + b) and Var(aX + b), and E(aX + bY) and Var(aX + bY) for independent X and Y. Exclude normal approximation to binomial distribution. The concept of continuous random variables is for teaching and learning only.
Lessons
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About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Discrete variables, expectation and varianceBuild probability distributions, check moments and transformations, and compare dependent and independent reward rules.
- Binomial models and calculator tailsChoose a binomial model, translate exact and conditional tails, recover parameters and combine component and box counts.
- Normal models and linear combinationsInterpret normal areas and inverse tails, recover parameters, and compare transformed or independent measurements under a model.