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2.4.4 Recap of Binomial Distribution

Key Characteristics of a Binomial Experiment

  1. Identical and Independent Trials: Each trial in the experiment is identical, meaning the probability of success remains constant across trials, and trials are independent of one another.
  2. Two Outcomes per Trial: Each trial has only two possible outcomes—success or failure.
  3. Fixed Number of Trials: The number of trials, denoted as n, is predetermined before the experiment begins.
  4. Consistent Probability of Success: The probability of success, denoted as p, does not change from one trial to the next.

Properties of the Binomial Random Variable

  1. Counts the Number of Successes: A binomial random variable sums up the total number of successes across n trials.
  2. Probability Mass Function (PMF): Given by the formula

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where: - k represents the number of successes, - n is the total number of trials, - p is the probability of success, - (1-p) is the probability of failure. 3. Mean and Variance: - Mean (Expected Value): E[Y]=np - Variance: Var(Y)=np(1−p)

When Binomial Conditions are Not Met

  • If the assumptions of identical trials or independence between trials are not met, the distribution may not be binomial.
  • In such cases, other probability distributions or models might be more appropriate.

Practical Application and Limitations

  • Binomial distributions model a wide range of real-world phenomena, such as coin tosses, quality control in manufacturing (like defect rates), and customer behavior (like conversion rates in marketing).
  • However, it's crucial to validate that the assumptions hold reasonably well in practice, or adjustments may be necessary.