07. Expectation & Conditional expectation of the normal distribution

Understanding Expectation & Conditional Expectation in Probability

Key Concepts:

• Expectation (Mean):

  • Represents the average outcome over all possible outcomes in a normal distribution.
  • Parametrized by Mu, calculated over the entire range (negative to positive infinity).

• Conditional Expectation:

  • Used when additional information is available (e.g., knowing someone’s IQ is above 110).
  • Allows for a more refined mean calculation based on given conditions.

Application Example:

• Scenario:

  • Evaluating the IQ of a person knowing it's above 110.
  • Uses the normal distribution (Mean: 100, SD: 15), typical for IQ scores.

• Calculation:

  • Integrate using a Lambda function from 110 to infinity to find expectation over this range (30.04).
  • Divide by the probability of having IQ > 110 (calculated using CDF).
  • Result: Adjusted IQ expectation is 118.98, indicative of the person's placement among higher IQ individuals.

Broader Implications:

- Calculating risk in trading strategies using conditional expectations.
- Expected shortfall as a risk measure.

Focus on understanding Brownian motion for further learning.