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• Probability Distributions

  A combination of the methods of descriptive statistics and those of probability to describe and analyze probability distributions

  
• Probability distributions

  Describe what will probably happen instead of what actually did happen, and they are often given in the format of a graph, table, or formula

  
• Random variable

  A variable (typically represented by X) that has a single numerical value, determined by chance, for each outcome of a procedure
• Discrete random variable

  Either a finite number of values or countable number of values, where "countable" refers to the fact that there might be infinitely many values, but that they result from a counting process
• Continuous random variable

  Has infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions

  
• Probability distribution

  • The sum of all probabilities must be 1
  • Each probability value must be between 0 and 1 inclusive

  
• Probability histogram

  Very similar to a relative frequency histogram, but the vertical scale shows probabilities

  
• Calculating mean, variance and standard deviation of a probability distribution

  1. Mean: E(x) = ∑[xP(x)]

  2. Variance: σ^2 = ∑[(x-μ)^2 P(x)]

  3. Standard deviation: σ = √(σ^2)

  
• According to the range rule of thumb, most values should lie within 2 standard deviations of the mean
• Unusually high/low values

  • Unusually high: x successes among n trials is unusually high if P(x or more) ≤ 0.05
  • Unusually low: x successes among n trials is unusually low if P(x or fewer) ≤ 0.05
• Binomial probability distributions

  • The procedure has a fixed number of trials
  • The trials must be independent
  • Each trial must have all outcomes classified into two categories (success and failure)
  • The probability of a success remains the same in all trials
• p and q

  p = probability of success, q = probability of failure (q = 1-p)
• Finding binomial probabilities

  1. Using the binomial probability formula: P(x) = (n!/(n-x)!x!)p^x q^(n-x)

  2. Using technology (mathematical/statistical software, spreadsheets, calculators)

  
• When sampling without replacement, consider events to be independent if n < 0.05N
• Binomial distribution parameters

  • Mean: μ = np
  • Variance: σ^2 = npq
  • Standard deviation: σ = √(npq)

  
• The maximum usual number of school leavers wanting to join UoM is 13. It is not unusual for everyone in the group to want to join UoM.
• Poisson probability distributions

  Used for describing the behaviour of rare events: events with relatively low probabilities of occurrence
• SIS 1037Y

  2020/2021
• 95% of school leavers want to join UoM. A group consists of 12 randomly selected school leavers.
• The max usual number of school leavers wanting to join UoM is 13. It is not unusual for everyone in the group to want to join UoM.
• Topics

  • Probability Distributions
  • Binomial Probability Distributions
  • Parameters for Binomial Distributions
  • Poisson Probability Distributions
  • The Standard Normal Distribution
  • Applications of Normal Distributions
  • Sampling Distributions and Estimators
  • Assessing Normality
  • Normal as Approximation to Binomial and Poisson

  
• Poisson distribution

  A discrete probability distribution which is often used for describing the behaviour of rare events: events with small probabilities
• Poisson distribution

  1. The random variable x is the number of occurrences of the event in an interval

  2. The interval can be time, distance, area, volume, or some similar unit

  3. P(x) = μxe-μ/x!

  
• Parameter λ

  Used instead of μ in Poisson distribution
• Poisson distribution

  • The random variable x is the number of occurrences of an event over some interval
  • The occurrences must be random
  • The occurrences must be independent of each other
  • The occurrences must be uniformly distributed over the interval being used

  
• Mean μ

  Mean number of occurrences of the event over the interval
• Variance σ2

  Equal to μ in Poisson distribution
• Standard deviation σ

  Equal to √μ in Poisson distribution
• Binomial distribution

  Affected by the sample size n and the probability p

  
• Poisson distribution

  Affected only by the mean μ
• In a binomial distribution the possible values of the random variable x are 0, 1, . . ., n, but a Poisson distribution has possible x values of 0, 1, 2, . . . , with no upper limit.

  
• Assuming a Poisson distribution as a suitable model for 530 cyclones over 100 years.
• Mean μ

  No. cyclone/no. years = 5.3
• P(2) = 5.32*e-5.3/2! = 0.0701
• P(0) = 5.30*e-5.3/0!
• P(1) = 5.31*e-5.3/1!
• The Poisson distribution is sometimes used to approximate the binomial distribution when n is large and p is small. The larger the n and the smaller the p, the better is the approximation.
• Rule of Thumb to Use the Poisson to Approximate the Binomial

  n ≥ 100, np 10
• The approximation is good when p < 0.05 and n > 20
• What about n > 40 and p < 0.1?