Cambridge A Level Probability & Statistics 2 (Paper 6)

• calculate probabilities for the distribution Po(μ) ;
• use the fact that if X ~ Po(μ) then the mean and variance of X are each equal to μ ;
• understand the relevance of the Poisson distribution to the distribution of random events, and use the Poisson distribution as a model;
• use the Poisson distribution as an approximation to the binomial distribution where appropriate (n > 50 and np < 5, approximately);
• use the normal distribution, with continuity correction, as an approximation to the Poisson distribution where appropriate ( μ > 15, approximately).

E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X) ,

E(aX + bY) = aE(X) + bE(Y),

Var(aX + bY) = a²Var(X) + b²Var(Y) for independent X and Y,

if X has a normal distribution then so does aX + b ,

if X and Y have independent normal distributions then aX + bY has a normal distribution,

if X and Y have independent Poisson distributions then X + Y has a Poisson distribution.

• understand the concept of a continuous random variable, and recall and use properties of a probability density function (restricted to functions defined over a single interval);
• use a probability density function to solve problems involving probabilities, and to calculate the mean and variance of a distribution (explicit knowledge of the cumulative distribution function is not included, but location of the median, for example, in simple cases by direct consideration of an area may be required).

• understand the distinction between a sample and a population, and appreciate the necessity for randomness in choosing samples;
• explain in simple terms why a given sampling method may be unsatisfactory (knowledge of particular sampling methods, such as quota or stratified sampling, is not required, but candidates should have an elementary understanding of the use of random numbers in producing random samples);
• recognise that a sample mean can be regarded as a random variable,and use the facts that \(E(\overline{X}) = \mu\) and that \(Var(\overline{X}) = \dfrac{\sigma^2}{n}\) ;
• use the fact that \(\overline{X}\) has a normal distribution if \(X\) has a normal distribution;
• use the Central Limit theorem where appropriate;
• calculate unbiased estimates of the population mean and variance from a sample, using either raw or summarised data (only a simple understanding of the term ‘unbiased’ is required);
• determine a confidence interval for a population mean in cases where the population is normally distributed with known variance or where a large sample is used;
• determine, from a large sample, an approximate confidence interval for a population proportion.

• understand the nature of a hypothesis test, the difference between one-tail and two-tail tests, and the terms null hypothesis, alternative hypothesis, significance level, rejection region (or critical region), acceptance region and test statistic;
• formulate hypotheses and carry out a hypothesis test in the context of a single observation from a population which has a binomial or Poisson distribution, using either direct evaluation of probabilities or a normal approximation, as appropriate;
• formulate hypotheses and carry out a hypothesis test concerning the population mean in cases where the population is normally distributed with known variance or where a large sample is used;
• understand the terms Type I error and Type II error in relation to hypothesis tests;
• calculate the probabilities of making Type I and Type II errors in specific situations involving tests based on a normal distribution or direct evaluation of binomial or Poisson probabilities.