Mathematics 9709 — specification explorer

1. The Poisson distribution

• 6.1.1

  use formulae to calculate probabilities for the distribution Po(λ):

  • Proofs are not required.
  2 resources
• 6.1.2

  use the fact that if X~Po(λ) then the mean and variance of X are each equal to λ:

  • Proofs are not required.
  2 resources
• 6.1.3understand the relevance of the Poisson distribution to the distribution of random events, and use the Poisson distribution as a model
  2 resources
• 6.1.4use the Poisson distribution as an approximation to the binomial distribution where appropriate
  2 resources
• 6.1.5use the normal distribution, with continuity correction, as an approximation to the Poisson distribution where appropriate.
  2 resources

2. Linear combinations of random variables

• 6.2.1

  use, when solving problems, the results that Var( aX + b ) = a Var( X ):

  • E( aX + b ) = aE ( X ) + b and
  • E( aX + bY ) = a E( X ) + b E( Y )
  3 resources
• 6.2.2

  2 independent X and Y aX + b distributions then aX + bY has a normal distribution distributions then X + Y has a Poisson distribution.:

  • Var(a X + b Y ) = a Var( X ) + b Var( Y ) for
  • if X has a normal distribution then so does
  • if X and Y have independent normal
  • if X and Y have independent Poisson
  3 resources

3. Continuous random variables

• 6.3.1understand the concept of a continuous random variable, and recall and use properties of a probability density function
  1 resource
• 6.3.2use a probability density function to solve problems involving probabilities, and to calculate the mean and variance of a distribution.
  1 resource

4. Sampling and estimation

• 6.4.1understand the distinction between a sample and a population, and appreciate the necessity for randomness in choosing samples
• 6.4.2explain in simple terms why a given sampling method may be unsatisfactory
• 6.4.3recognise that a sample mean can be regarded as a random variable, and use the facts that
• 6.4.4

  E X _ i = n and that Var _ X i = X has a normal distribution Theorem (CLT) is required; for large sample sizes, the distribution of a sample mean is approximately normal. mean and variance from a sample, using either required, e.g. that although individual estimates will raw or summarised data vary the process gives an accurate result ‘on average’. a population mean in cases where the population is normally distributed with known variance or where a large sample is used confidence interval for a population proportion.:

  • use the fact that _ X i has a normal distribution if
  • use the Central Limit Theorem where appropriate Only an informal understanding of the Central Limit
  • calculate unbiased estimates of the population Only a simple understanding of the term ‘unbiased’ is
  • determine and interpret a confidence interval for
  • determine, from a large sample, an approximate
  3 resources

5. Hypothesis tests

• 6.5.1understand the nature of a hypothesis test, the difference between one-tailed and two-tailed tests, and the terms null hypothesis, alternative hypothesis, significance level, rejection region (or critical region), acceptance region and test statistic
• 6.5.2

  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 the Poisson distribution, where appropriate:

  • direct evaluation of probabilities
  • a normal approximation to the binomial or
• 6.5.3formulate 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
• 6.5.4understand the terms Type I error and Type II error in relation to hypothesis tests
• 6.5.5calculate 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.
  4 resources