Probability

• Random Variable


  A variable whose possible values are numerical outcomes of a random experiment

  
• Random Variable

  • A real valued function defined on the sample space of a random experiment
  • There are two types: discrete and continuous

  
• Discrete Random Variable


  A random variable which may take on only a countable number of distinct values

  
• Examples of discrete random variables


  • Number of children in a family
  • Friday night attendance at a cinema
  • Number of patients in a doctor's surgery
  • Number of defective light bulbs in a box of ten
• Probability distribution


  A summary of probabilities for the values of a random variable

  
• Probability distribution of a discrete random variable


  A list of probabilities associated with each of its possible values

  
• Cumulative distribution function


  A function that gives the probability that the random variable is less than or equal to a given value

  
• Computation of Cumulative Distribution Function

  Steps to compute the cumulative distribution function
• Bernoulli Random Variable


  A random variable that can only take one of two values - one and zero, where one means the experiment was a success and zero means it failed
• Bernoulli Random Variable


  • It has only one independent trial (one flip of coin, or one roll of a die)
  • It can only take one of two values - one and zero
• Binomial Random Variable


  Represents the number of successes in a fixed number of successive identical, independent trials, where each trial has the possibility of either two outcomes: Success or Failure
• Binomial Random Variable


  • The probability of the two outcomes remains constant for every trial
• Probability Mass Function (PMF)

  Formula to calculate the probability mass function

  
• Bernoulli Random Variable


  The most straightforward kind of a random variable
• Binomial Random Variable


  • A random variable follows binomial distribution if: 1) There are a fixed number of trials 2) Each trial has only two possible outcomes (success or failure) 3) The probability of success remains constant for each trial
• Properties


  Mean = np

  Variance = npq

  
• Charlie has a 0.0923 chance of making precisely 4 out of the next seven free throws.
• Charlie is expected to make about 5.74 free throws out of 7 shots, with a standard error of 1.016.
• Poisson Probability Distribution


  A random variable follows a Poisson probability distribution if: 1) Events occur at random in a given time period 2) The average rate of events is 'a' per time period 3) The time period for the rate is the same as the time period for counting events
• Examples of Poisson random variables

  • Number of breakdowns of a machine in an 8-hour shift

  Number of cars arriving at a parking garage in 1 hour

  Number of sales made by a telesales person in a week
• Poisson Probability Mass Function

  Formula to calculate the probability of x occurrences of an outcome in a predetermined time, space or volume interval

  
• Mean and Variance of Poisson Distribution

  Mean = a

  Variance = a
• Continuous Random Variable


  A random variable that can take on any value (as opposed to only discrete values) in an interval

  
• Examples of continuous random variables

  • Length of time to complete a task

  Mass of a passenger's hand luggage

  Daily distance travelled by a delivery vehicle

  Volume of fuel in a tank
• Probability Density Function


  A function that describes the relative likelihood for a random variable to occur at a given point

  
• Continuous Probability Distributions


  • Represented by curves where the area under the curve between two x-limits represents the probability that x lies within these limits

  The normal distribution is the most widely used continuous probability distribution

  
• Normal Distribution

  • Bell-shaped curve

  Symmetrical about the mean value μ

  Tails never touch the x-axis (asymptotic)

  Described by two parameters: mean (μ) and standard deviation (σ)

  Total area under the curve equals 1

  
• Continuous random variable

  An infinite number of possible outcomes, represented by curves where the area under the curve between two x-limits represents the probability that x lies within these limits

  
• Normal distribution


  • A large majority of continuous random variables have outcomes that follow a normal pattern
  • The curve is bell-shaped
  • It is symmetrical about a central mean value, μ
  • The tails of the curve never touch the x-axis, meaning that there is always a non-zero probability associated with every value in the problem domain (i.e. asymptotic)
  • The distribution is always described by two parameters: a mean (μ) and a standard deviation (σ)
  • The total area under the curve will always equal one, since it represents the total sample space
  • The area under the curve below μ is 0.5, and above μ is also 0.5

  
• Finding probabilities using the normal distribution

  1. Convert x-limits into limits that correspond to the standard normal distribution (z-distribution)

  2. Use the z-table to find the probability that x lies between the limits

  
• Standard normal (z) distribution

  • Has a mean of 0 and a standard deviation of 1
  • Probabilities (areas) based on the standard normal distribution can be read from standard normal tables (z-table)
• Reading values (areas) from the z-table


  1. The z-limit (to one decimal place) is listed down the left column, and the second decimal position of z is shown across the top row

  2. The value read off at the intersection of the z-limit (to two decimal places) is the area under the standard normal curve (i.e. probability) between 0 < z < k
• Determining probabilities using the z-table


  • P(0 < z < 1.46)
  • P(-2.3 < z < 0)
  • P(z > 1.82)
  • P(-2.1 < z < 1.32)
  • P(1.24 < z <2.075)
• Finding probabilities for x-limits using the z-distribution


  1. Convert each x-value into a z-value using the formula: z = (x - μ) / σ

  2. Use the z-table to find the probability associated with the z-value range

  
• Interpretation of z-value


  Measures how far (in standard deviation terms) an x-value lies from its mean, μ

  
• Courier service delivery time example

  • Delivery time is normally distributed with μ = 45 minutes and σ = 8 minutes
  • Find the probability a randomly selected parcel will take between 45 and 51 minutes to deliver
  • Find the probability a randomly selected parcel will take less than 48 minutes to deliver
• Finding x-limits for given probabilities of a normal distribution

  1. Sketch the normal curve and show the position of the x-value that corresponds to the given probability

  2. Use the z-table to identify the z-value that corresponds to the area under the normal curve

  3. Convert the z-value into its corresponding x-value using the z transformation formula

  
• Clothing store transaction value example

  • Transaction value is normally distributed with μ = R244 and σ = R68
  • Find the minimum purchase value for the highest-spending 15% of customers
  • Find the purchase value that separates the lowest-spending 20% of customers from the remaining customers
• If A and B are independent events, then P(A and B) = P(A)*P(B).
• If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B).