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6. Probability & Statistics 2

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  1. 6.1 The Poisson distribution

    1. 6.1.1Poisson probabilities

      • use formulae to calculate probabilities for the distribution Po m^h

    2. 6.1.2The fact that if X Po+

      • use the fact that if X Po+ m^h then the mean and variance of X are each equal to m Proofs are not required.

    3. 6.1.3Poisson distribution

      • understand the relevance of the Poisson distribution to the distribution of random events, and use the Poisson distribution as a model

    4. 6.1.4Binomial distribution

      • use the Poisson distribution as an approximation to the binomial distribution where appropriate The conditions that n is large and p is small should be known; n > 50 and np < 5, approximately.

    5. 6.1.5Normal distribution

      • use the normal distribution, with continuity correction, as an approximation to the Poisson distribution where appropriate. The condition that m is large should be known; 152m, approximately.

  2. 6.2 Linear combinations of random variables

    1. 6.2.1

      • use, when solving problems, the results that - E(aX + b) = aE(X) + b and Var(aX + b) = a2 Var(X) - E(aX + bY) = aE(X) + bE(Y) - Var(aX + bY) = a2 Var(X) + b2 Var(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. Proofs of these results are not required.

  3. 6.3 Continuous random variables

    1. 6.3.1Continuous random variables

      • understand the concept of a continuous random variable, and recall and use properties of a probability density function For density functions defined over a single interval only; the domain may be infinite, e.g. x 3 4 for x 1H.

    2. 6.3.2Density functions

      • use a probability density function to solve problems involving probabilities, and to calculate the mean and variance of a distribution. Including location of the median or other percentiles of a distribution by direct consideration of an area using the density function. Explicit knowledge of the cumulative distribution function is not included.

  4. 6.4 Sampling and estimation

    1. 6.4.1Samples and populations

      • understand the distinction between a sample and a population, and appreciate the necessity for randomness in choosing samples

    2. 6.4.2Sampling methods

      • explain in simple terms why a given sampling method may be unsatisfactory Including an elementary understanding of the use of random numbers in producing random samples. Knowledge of particular sampling methods, such as quota or stratified sampling, is not required.

    3. 6.4.3Sample mean

      • recognise that a sample mean can be regarded as a random variable, and use the facts that E X = n_ i and that Var X n 2 = v_ i

    4. 6.4.4Normal distribution

      • use the fact that X_ i has a normal distribution if X has a normal distribution

    5. 6.4.5Sample mean

      • use the Central Limit Theorem where appropriate Only an informal understanding of the Central Limit Theorem (CLT) is required; for large sample sizes, the distribution of a sample mean is approximately normal.

    6. 6.4.6Unbiased estimates

      • 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, e.g. that although individual estimates will vary the process gives an accurate result 'on average'.

    7. 6.4.7Mean confidence intervals

      • determine and interpret 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

    8. 6.4.8Proportion confidence intervals

      • determine, from a large sample, an approximate confidence interval for a population proportion.

  5. 6.5 Hypothesis tests

    1. 6.5.1Hypothesis tests

      • understand 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 Outcomes of hypothesis tests are expected to be interpreted in terms of the contexts in which questions are set.

    2. 6.5.2Normal approximation

      • 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 - direct evaluation of probabilities - a normal approximation to the binomial or the Poisson distribution, where appropriate

    3. 6.5.3Hypothesis tests

      • 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

    4. 6.5.4Hypothesis tests

      • understand the terms Type I error and Type II error in relation to hypothesis tests

    5. 6.5.5Normal distribution

      • 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.