A Level Math statistics 2

1. The Poisson distribution

Poisson Distribution

• Introduction
• Changing the mean
• Finding an observed value x
• Exam Questions

Continuity Corrections

• Continuity Corrections

Poisson Approximation to the Binomial Distribution

• Poisson approximation to the Binomial distribution
  • An example for you to try
• Exam Questions

Normal Approximation to the Poisson Distribution

• The normal approximation to the Poisson distribution
• Exam Questions

2. Linear combinations of random variables

• Linear combinations of random variables
• use, in the course of solving problems, the results that
  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.

3. Continuous random variables

Probability Density Functions and Cumulative Distribution Functions

• What is a Probability Density Function (p.d.f,)?
  • Finding the constant k in a p.d.f
  • Calculating probability from a p.d.f. (this is over two intervals, you are only expected to go over a single interval but it shold still help)
  • The cumulative distribution function, (c.d.f) (not necessarily expected)
    • Harder example on the c.d.f. function
  • Finding the median from a p.d.f.
    • An example for you to try
  • Finding the mode from a p.d.f.
  • E(X) and Var(X)
    • An example for you to try
• Exam Questions

4. Sampling and estimation

• Simple random sampling
• What is a statistic?
• Sampling distribution of the sample mean
• Sampling distribution of the mode and median
• Exam Questions
• 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(sample mean ) = μ and that Var(sample mean)= σ²/n
• use the fact that the sample mean 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.

5. Hypothesis tests

Binomial Distribution

• Introduction
  • Lower tail test
  • Upper tail test
  • Critical values - lower tail test
  • Critical values - upper tail test
  • Two tailed test
• Exam Questions

Poisson Distribution

• Test for a Poisson mean
  • Lower tail test
  • Upper tail test
  • Two tailed test
• Exam Questions

• 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.