STATS Module 5

• Random variable


  A variable whose value is determined by the outcome of a random experiment
• Random variable examples


  • How many customers will visit a bank, a grocery store, or a gas station on a given day?
  • How many cars will pass a bridge on a given day?
  • How many students will be absent from a class on a given day?
• Random variable


  • Each can assume any value within a certain interval
• Types of variables


  • Discrete variable
  • Continuous variable
• Discrete variable


  Assume countable values
• Continuous variable


  Assume any value contained in one or more intervals
• Discrete random variable examples


  • Count cars at toll between 11:00 & 1:00
  • Make 100 sales calls
  • Inspect 70 radios
  • Answer 33 questions

  
• Continuous random variable examples


  • Time taken to commute from home to work
  • Amount of milk in a gallon
  • Weight of a letter
  • Price of a house
• Probability distribution of a discrete random variable

  A graph, table, or formula that specifies the probability associated with each possible value the random variable can assume

  
• Discrete probability distribution example

  • Toss 2 coins, count number of tails
• Probability distribution of a discrete random variable possesses two characteristics:
• Steps to find probabilities from a discrete probability distribution

  1. P(selected family owns two vehicles)

  2. P(selected family owns at least two vehicles)

  3. P(selected family owns at most one vehicle)

  4. P(selected family owns three or more vehicles)

  
• A valid probability distribution table must have the sum of all probabilities equal to 1.0

  
• Probability distribution table examples

  • Table 1: Invalid

  Table 2: Valid

  Table 3: Invalid

  
• Steps to find probabilities from a discrete probability distribution

  1. P(exactly 2 breakdowns)

  2. P(0 to 2 breakdowns)

  3. P(more than 1 breakdown)

  4. P(at most 1 breakdown)

  
• Mean of a discrete random variable


  The value that is expected to occur per repetition, on average, if an experiment is repeated a large number of times
• Standard deviation of a discrete random variable

  Measures the spread of its probability distribution

  
• Binomial probability distribution is one of the most widely used discrete probability distributions
• Binomial experiment


  • There are n identical trials

  2. Each trial has two and only two outcomes (success and failure)

  3. The probability of success is p and that of failure is q, where p + q = 1

  4. The trials are independent
• Binomial probability formula


  P(x) = nCx * p^x * (1-p)^(n-x)

  
• Binomial probability distribution example


  • Toss 1 coin 5 times, find probability of 3 tails
• Binomial experiment


  An experiment that satisfies the following four conditions:

  1. There are a fixed number of identical trials

  2. Each trial has two possible outcomes: success or failure

  3. The probability of success is constant for each trial

  4. The trials are independent
• Binomial probability formula


  P(x) = (n choose x) * p^x * (1-p)^(n-x)

  where:

  n = total number of trials

  x = number of successes

  p = probability of success

  q = 1-p = probability of failure
• The binomial probability distribution is symmetric if p=.50, skewed to the right if p<.50, and skewed to the left if p>.50

  
• The probability that all 4 students selected are undergraduate students is 0.062