CIE A Level Maths: Probability & Statistics 1复习笔记3.2.3 The Geometric Distribution

What is a geometric distribution and its notation?

• A geometric distribution is a discrete probability distribution
• 
  • Each trial has only two outcomes
    Broadly labelled as “success” and “failure” - these are mutually exclusive

Such trials may be referred to as Bernoulli trials – named after the Swiss mathematician Jacob Bernoulli (1655-1705)

• • The outcomes of trials are independent
    The outcome of one trial does not affect the outcome of another trial
  • The probability of each outcome is constant across all trials

i.e.   The probability of “success” does not change between trials

• p is the probability of “success” in a single trial
  • The probability of “failure” in a single trial is 1- p ; often denoted by q

What does a geometric distribution look like?

• If represented visually, using a vertical line graph, the probabilities in a geometric distribution decrease but never reach zero
  • The probabilities form a geometric progression
    •                        (Answer below diagram)
    • The probabilities decrease exponentially; drawing a curve through the tops of the lines would produce a decreasing exponential curve
  • The graphs also show how 1 is the mode for every geometric distribution

What are the properties of a geometric distribution?

How do I set up a geometric model?

• Identify what a trial is in the context of a problem
  • Flipping a coin, rolling a dice, a football match
• Identify a successful outcome
  • Heads, a square number, a win
• Identify the probability of “success”

What can be modelled using a geometric distribution?

• Anything where the first occurrence of a successful outcome is of significance
  • Rolling a double with two dice before being allowed to start a game
  • The number of on/off presses a switch can withstand before wearing out

(In which case the first “success” would be the first failure of the switch!)

• In addition, the scenario must satisfy the three conditions
  • Trials only have two outcomes of interest
  • Trials are independent
  • Probability of success is constant for all trials
  • These are also three of the four conditions for a binomial distribution so are not enough on their own – it will also depend on the context
• Many scenarios may appear as having more than two outcomes but in the context of the question only two are of significance
  • e.g. A light that randomly flashes in 8 different colours, but the only colour of interest is blue
    So “blue” is “success” and all other colours, regardless of whether it  is red, yellow, etc – i.e. “not blue” - is “failure”
• Sometimes a sample may be taken from a population
  • As long as the population is large enough and the sample is random the probability of “success” in the sample is the same as the probability of “success” in the population

What cannot be modelled using a geometric distribution?

• Be careful not to confuse binomial and geometric distributions/models
  • Binomial is for the number of successes in a fixed number of trials
  • Geometric is for the number of trials up to and including the first success
• Anything where a trial would have more than two outcomes of interest
  • e.g. Outcome of a football match – win, draw or lose
• Where the probability of an outcome of a trial is influenced by a previous trial
  • i.e. trials are not independent
  • e.g. drawing counters from a bag without replacement
• Anything where the probability of “success” changes with time – or practice
  • e.g. a skateboarder performing a trick - the probability of success should increase after practising the trick

How do I calculate geometric probabilities?

• Identify p, the probability of “success” and 1- p, the probability of “failure” (q)
• The mode is 1 for all geometric distributions