ECON1006 Introduction to Economic Methods - Week 4 Summary

Required Reading

• Ref. File 4: Sections 4.7 to 4.9
• Ref. File 5: Introduction and Sections 5.1 to 5.4

4. Probability Theory Continued

4.9 Sampling With and Without Replacement

• Random Sample Definition: A sample where every possible combination of elements has an equal probability of selection.
• Sampling with Replacement:
  • Observations are returned to the population before next selection.
  • Population remains constant and selections are independent.
• Sampling without Replacement:
  • Population decreases with each selection.
  • Each selection outcome depends on previous outcomes.
• Examples:
  • Probability calculations around voting and card drawing scenarios illustrate sampling differences.

4.10 Probability Trees

• Purpose: Helps calculate joint probabilities (probabilities of intersections of events).
• Example: Greasy Mo’s meal deal probabilities using tree diagrams to calculate the likelihood of selecting a pizza and fruit juice.

5. Probability Distributions of Discrete Random Variables

5.1 Probability Distributions and Random Variables

• Definition: A model representing the relative frequency distribution of outcomes.
• Random Variable:
  • Assigns numerical values to outcomes of a statistical experiment.
  • Can be discrete (finite or countable values) or continuous (any value in an interval).
• Examples: Hair color or number of children can be represented as random variables.

5.2 Expected Values of Random Variables

• Expected Value:
  • A measure of the center of the probability distribution of a random variable.
  • Calculated as a weighted average of all possible values.
• Properties: Includes various properties and theorems related to expected values.
• Example: Calculating expected net gain from a lottery.

5.3 The Variance of a Random Variable

• Variance:
  • Measures dispersion of a random variable around its mean.
  • Defined for both discrete and continuous variables.
• Standard Deviation: Square root of variance.
• Example: Calculating variance and standard deviation for lottery prizes.

5.4 The Binomial Distribution

• Bernoulli Experiments:
  • Two possible outcomes: success or failure.
  • Example: Coin toss with head as success.
• Binomial Experiments:
  • Involves 'n' trials of a Bernoulli experiment.
  • Independent trials with constant probability of success.
• Binomial Probability Function:
  • Formula to calculate probabilities of 'x' successes in 'n' trials.
  • Examples provided with calculations for air conditioning units requiring service.
• Cumulative Binomial Probabilities:
  • Often calculated with tables due to tedious nature.
• Characteristics:
  • Mean and variance formulas for binomial random variables.
  • Skewness depends on the relationship between 'p' and '0.5'.

Main Points

• Sampling without replacement depends on previous outcomes.
• Sampling with replacement is like sampling from an infinite population.
• Tree diagrams aid in joint probability calculations.
• Probability distributions model statistical populations and may approximate real distributions.
• Random variables help frame probability distributions.
• Binomial distribution models relative frequency of successes in Bernoulli trials.
• Binomial distribution probability function depends on number of trials, successes, and success probability.