Introduction to Random Variables and Probability Models

• Insurance companies use probability models to estimate risk and determine policy pricing.

• The model helps calculate fair prices considering possible payouts and associated probabilities.

Random Variables

• Random Variable, X: Value based on random event outcome.

  • Discrete Random Variable: Has a finite list of outcomes.

  • Continuous Random Variable: Can take any value (real numbers, possibly bounded).

  • Probability Distribution: Collection of all possible values of a random variable with their probabilities.

  • Probabilities can be listed in a table (discrete) or described by a function (continuous).

Properties of Discrete Probability Models

• Lists all possible outcomes.

• Matches each outcome with a probability.

• Each probability is between 0 and 1.

• The total of all probabilities equals 1.

Expected Value of a Random Variable

• Theoretical expected value is the average outcome based on probabilities.

• Noted as E(X).

• Calculated with E(X) = sum of (outcome x probability).

Example: Insurance Policy

• Death rate: 1 out of 1000. Disability: 2 out of 1000.

• Expected Value from policy can be calculated based on these rates.

Standard Deviation of a Random Variable

• The measure of variability in the outcomes, typically involves:

  • Individual payouts, probabilities, deviations from mean, and squared deviations weighted by probabilities.

Baby Shark Tank Example

• A financial institution evaluates a loan with profits based on product success:

  • Success Levels: Low (0.1), Moderate (0.3), High (0.6).

  • Expected Values:

  • Full loan: $33.5 thousand ($33,500).

  • Half loan: $14 thousand ($14,000).