Difference between Expectation and Average

What’s the Difference Between Expectation and Average?

• Expectations (Expected Value): In probability theory, the expectation is the theoretical mean of a random variable over an infinite number of trials. It’s calculated as the sum of all possible values each multiplied by its probability of occurrence.
• Average (Arithmetic Mean): In statistics, the average is the sum of observed values divided by the number of observations. It’s an empirical measure of central tendency based on actual data.

Key Definitions

• Expectation (Expected Value):

  • Random Variable: A variable whose possible values are numerical outcomes of a random phenomenon.
  • Probability Distribution: A function that describes the likelihood of obtaining the possible values that a random variable can assume.

• Average (Arithmetic Mean):

  • Sample Data: A set of data collected from a population by a defined procedure.
  • Central Tendency: A central or typical value for a probability distribution.

Examples

• Expectation: The expected value of rolling a fair six-sided die is calculated as $E[X]=(1\times 1/6)+(2\times 1/6)+(3\times 1/6)+(4\times 1/6)+(5\times 1/6)+(6\times 1/6)=3.5$.

• Average: If you roll a die 5 times and get the results $2,3,5,6,4$, the average is $(2+3+5+6+4)/5=4$

• Linking Both: As the number of observations increases, the average of the observed data tends to approach the expected value due to the Law of Large Numbers.

Critical Thinking Analysis

• Thesis:

  • FACT: Expectation is a theoretical measure calculated using the probability distribution of a random variable, considering all possible outcomes and their probabilities.
  • REASONING: It provides a prediction of the long-term average outcome of a random process, essential for probabilistic modeling.

• Antithesis:

  • FACT: Average is an empirical measure derived from actual data collected from experiments or observations.
  • REASONING: It summarizes observed data but may not accurately reflect the underlying probability distribution, especially with limited or biased samples.

• Synthesis:

  • FACT: Both expectation and average aim to quantify central tendency but differ in their approaches—one is theoretical, and the other is empirical.

  • REASONING: Understanding their differences is crucial for interpreting statistical results and making accurate predictions based on data.

Literature Review

Weiss, N.A. (2018)

• Introductory Statistics (10th Edition), Weiss, N.A., 2018

• Key Points:

  • Differentiates between population parameters (expectation) and sample statistics (average).
  • Explains how sample means converge to expected values with larger sample sizes.
  • Emphasizes the importance of understanding the underlying distribution when interpreting averages.

Ross, S.M. (2014)

• Introduction to Probability Models (11th Edition), Ross, S.M., 2014
• Key Points:
  • Provides a comprehensive explanation of expected value and its properties.
  • Discusses the distinction between expected value and sample mean in probabilistic terms.
  • Explores practical applications where expectation plays a critical role in modeling.

Related Concepts

• Law of Large NumbersProbabilityTheory

  • FACT: States that as the number of trials increases, the sample average converges to the expected value.
  • REASONING: It explains why averages from large samples are good estimates of the expected value.

• VarianceStatistics

  • FACT: Measures the dispersion of a set of values relative to their mean (expected value).
  • REASONING: Variance provides insight into the reliability of the expected value as a measure of central tendency.

• Central Limit TheoremProbabilityTheory

  • FACT: Indicates that the distribution of sample means approximates a normal distribution as the sample size becomes large.
  • REASONING: It justifies the use of normal probability models in statistical inference, linking sample averages to expectations.