Probability and Statistics 2 - Paper 6

Worked examples (14)extended set

• 1Strict and inclusive inequalities
• 2At least k, after rescaling the rate
• 3Stating and criticising a Poisson model
• 4Two types of sale: sum, split by type, conditional
• 5Why 2X is not Poisson
• 6Poisson approximation to a binomial, then a sum
• 7Show the approximation agrees to 2 d.p. but not 3 d.p.
• 8Finding λ from an equation (quadratic)
• 9Expressing one mean in terms of another (logs)
• 10Normal approximation with continuity correction
• 11Total of many values; smallest k
• 12Conditional probability with Poisson variables
• 13Finding the smallest n with a normal approximation
• 14Binomial to Poisson to normal: a full chain

Worked examples (13)extended set

• 1Converting units: E(aX + b) and Var(aX + b)
• 2Which runner is faster? P(X > Y) for two normals
• 3Mixed normal and Poisson: standard deviation only
• 4A score from a probability table: one round scaled vs many rounds
• 5Three rods vs three times one rod
• 6Two times that differ by more than 3 minutes
• 7Two Poisson streams combined, then a conditional
• 8Is it Poisson? X₁ + X₂ versus 2X
• 9Total mass of bags, then bags packed in a crate
• 10Fuel used over five days: show the variance, hence the probability
• 11Largest number of people for a lift
• 12Finding an unknown coefficient from a probability
• 13A large Poisson total approximated by a normal

Worked examples (12)extended set

• 1Show the constant, then find a probability
• 2Triangular pdf given as a graph
• 3Symmetric pdf — E(X) without calculation
• 4Straight-line pdf with unknown heights, then Var(X)
• 5Exact median from a quadratic, then a hence part
• 6Quartiles and interquartile range in context
• 7Pdf on an infinite domain
• 8Logarithmic pdf with exact answers
• 9Pdf with a parameter — probability independent of a
• 10Trigonometric pdf: symmetric interval and variance
• 11Exponential lifetime model linked to the binomial
• 12Mean and variance from a pdf, then the central limit theorem

Worked examples (14)extended set

• 1Unbiased estimates from raw data
• 2Show that the unbiased variance is a given value (coded data)
• 3Random numbers, a biased sample and a fair dice method
• 4Working backwards to Σx and Σx², then adding a value
• 5Sample mean from a normal population and the smallest sample size
• 6CLT for the mean of a binomial population
• 7Estimates, then a confidence interval using a given true variance
• 8Large-sample interval, interpreting it against a claim
• 9From the end-points back to x̄ and σ, then a new confidence level
• 10A 94% interval for a proportion and a claim
• 11Comparing widths to find an unknown confidence level
• 12From the width of a proportion interval to the possible counts
• 13What a confidence interval means: repeated intervals
• 14Sample mean from a discrete distribution: find k

Worked examples (13)extended set

• 1One-tailed binomial test by direct evaluation
• 2Test of a mean with known standard deviation
• 3Smallest significance level and errors in context
• 4Large-sample two-tailed test from summary data
• 5Two-tailed binomial test and its critical region
• 6Critical region at a level not in the tables
• 7Poisson critical value, Type I and Type II
• 8Binomial test using a normal approximation
• 9Poisson test using a normal approximation
• 10Reverse normal tests: critical sample mean, sample size and σ
• 11Test defined by a rule: significance level and Type II
• 12Normal test at 2%, then the full Type II method
• 13Type II error for a discrete population using the CLT