A-level Maths Statistics → Further Statistics 1

You can already

Use simple discrete probability distributions and calculate probabilities, including with the binomial model; A-level Maths deliberately excludes their mean and variance.

Now you'll

Calculate $E(X)$ and $\operatorname{Var}(X)$ for a discrete distribution, and extend expectation to functions such as $E(g(X))$.

You can already

Calculate binomial probabilities and decide whether a binomial model is appropriate.

Now you'll

Add the Poisson distribution, combine independent Poisson variables, find the mean and variance of both models, and use Poisson as an approximation to binomial.

You can already

Use the binomial model for the number of successes in a fixed number of independent trials.

Now you'll

Model the waiting time to the first success or to a later success with geometric and negative binomial distributions, including their means and variances.

You can already

Test a binomial proportion and a Normal mean using null and alternative hypotheses, critical regions and $p$-values.

Now you'll

Extend the same framework to tests for a Poisson mean and for the parameter $p$ of a geometric distribution.

You can already

Use the Normal distribution as a model and choose between the binomial and Normal models in context.

Now you'll

Use the Central Limit Theorem to approximate totals and means arising from other distributions.

You can already

Set up and interpret one- and two-tailed hypothesis tests for a single population parameter.

Now you'll

Carry out chi-squared goodness-of-fit tests and tests using contingency tables, including choosing the degrees of freedom.

You can already

List the probabilities in a discrete distribution, but without a function that encodes the whole distribution.

Now you'll

Define and use probability generating functions for binomial, Poisson, geometric and negative binomial variables; extract means and variances and handle sums of independent variables.

You can already

Interpret the significance level as the probability of rejecting a true null hypothesis.

Now you'll

Name and analyse Type I and Type II errors, the size and power of a test, and its power function.