3/11/25

Homework Review

• Question number four from discrete distributions homework discussed.

Overview of Life Insurance Example

• A 40-year-old man in the U.S. has a 0.242% risk of dying in the next year.

• Insurance company charges $250 per year for a life insurance policy that pays $100,000 on death.

• The profit for the insurance company depends on the outcome of the year, classified into two scenarios:

  • If you live: Profit = $250 (premium paid).

  • If you die: Profit = $100,000 - $250 = $99,750.

Probability Calculations

• Calculating Probabilities:

  • Probability of dying ( x = 1) = 0.242% = 0.00242.

  • Probability of living = 1 - 0.00242 = 0.99758.

Expected Value (Mean) Calculation

• Calculated in the context of the life insurance profits:

  • Mean = Sum of (Value * Probability).

• Given values and their probabilities for the table:

  • 4 * 0.12 = 0.48

  • 3 * 0.45 = 1.35

  • 2 * 0.33 = 0.66

  • 1 * 0.07 = 0.07

  • 0 * 0.03 = 0.00

• The total mean is estimated to be around 2.56.

Variance Calculation

• To find variance, the following is computed:

  • Formula: (Value - Mean)² * Probability for each value.

  • Example for four:

    • (4 - 2.56)² * 0.12 = 0.248832

  • Continue for values of 3, 2, 1, and 0.

• Variance result estimated at 0.8064.

  • Standard Deviation = √Variance = 0.898.

Introduction to Binomial Distribution

• Definition of binomial distribution details:

  • Fixed number of trials (n).

  • Two outcomes (success/failure).

  • Independent trials with constant probability of success (p).

  • Focus on counting the number of successes (X).

Notation for Binomial Distribution

• • Number of trials: n.

  • Probability of success: p (e.g., 0.6 for 60%).

  • Random variable (success count): X.

• Example of drawing cards (success = heart):

  • n = 10 trials, p = 0.25 for picking a heart, with x-values ranging from 0 to 10.

Probability Calculation Example

• Example scenario with trials of a 3-part experiment:

  • Probability of success of p = 0.4, two successes across three trials.

  • Combinations calculated through C(n, x) = C(3, 2) = 3 ways.

  • Probability for each sequence leading to say, success, failure, success.

  • General formula for binomial probabilities identified:

  • P(X = k) = C(n, k) * p^k * (1-p)^(n-k).

Actual Problem Example

• Multiple choice quiz scenario:

  • 10 questions, 5 choices each, chance to guess correct = 0.2.

  • To find probability of guessing 6 correct answers:

  • Use the binomial formula to evaluate:

  • C(10, 6) * 0.2^6 * 0.8^4.

  • Importance of calculator and correct inputs highlighted.

Calculator Use and Binomial CDF

• CDF used for non-exact probabilities (e.g., less than/equal to, greater than/equal to).

• Key calculations involve understanding setup and using correct calculator functionality:

  • Example questions were solved demonstrating the nuances of inputs required.

• Emphasis on subtle differences like whether to use PDF (specific) or CDF (cumulative).

Summary of Topics Covered

• Reviewed life insurance case for discrete probabilities.

• Introduced binomial distributions and their specific criteria.

• Provided numerous examples and probability calculations using specific formulas.

• Emphasized importance of calculators and practice with real problems.