Statistical Distributions

• Random variable


  A variable that can take any of a range of specific values, where the outcome is not known until the experiment is carried out
• Probability distribution


  Fully describes the probability of any outcome in the sample space
• Discrete uniform distribution


  When all probabilities are the same
• Discrete uniform distribution


  • Score when a fair dice is rolled
• Probability distribution of a discrete random variable


  1. Described using probability mass function

  2. Described using a table

  3. Described using a diagram
• Cumulative probabilities


  • P(X ≤ x) - Sum of all individual probabilities for values up to and including x
  • P(X < x) - Sum of all individual probabilities for values not greater than x
  • P(X ≥ x) - Sum of all individual probabilities for x and values greater than x
  • P(X > x) - Sum of all individual probabilities for values greater than x
• Binomial distribution


  Represents the number of successful trials when carrying out a number of trials
• Binomial distribution B(n, p)


  • There are a fixed number of trials, n
  • There are two possible outcomes (success and failure)
  • There is a fixed probability of success, p
  • The trials are independent of each other
• Binomial probability mass function


  P(X = r) = (n!/(r!(n-r)!)) * p^r * (1-p)^(n-r)
• You can use the binomial probability distribution function in the calculator to work out binomial probabilities
• Index


  n in the binomial distribution
• Parameter


  p in the binomial distribution
• Modelling a random variable X with a binomial distribution


  1. Define X to represent the number of successful trials

  2. Check the 4 conditions for a binomial distribution are met
• Binomial distribution example


  • Probability that a randomly chosen member of a reading group is left-handed is 0.15
  • A random sample of 20 members is taken
  • X = number of left-handed members in the sample
• The suitable model for X is X ~ B(20, 0.15)
• Calculating binomial probabilities


  1. Use the binomial probability mass function formula

  2. Use the binomial probability distribution function on the calculator
• The formula for the binomial coefficient is n! / (r! * (n-r)!)