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2.4.1 Bernoulli Random Variable and its Distribution

Definition

  • Bernoulli Random Variable: A discrete random variable that takes only two possible outcomes, typically coded as 1 (success) and 0 (failure).
  • Probability Mass Function (PMF): f(y)=p if y=1 and f(y)=1−p if y=0, where p is the probability of success and q = 1−p is the probability of failure.

Properties

image


Examples

  1. Coin Toss:
  2. Success (Heads) = 1 with probability p
  3. Failure (Tails) = 0 with probability 1-p

  4. Sales Call:

  5. Success (Sale made) = 1 with probability p
  6. Failure (No sale) = 0 with probability 1-p
  7. Customer Survey:
  8. Success (Satisfied) = 1 with probability p
  9. Failure (Not satisfied) = 0 with probability 1-p

  10. Store Demand:

  11. Success (Demand > 10 units) = 1 with probability p
  12. Failure (Demand ≤ 10 units) = 0 with probability 1-p

Application

  • Die Toss:
    Define success for getting a number less than or equal to 2 on a fair six-sided die: image

Calculation Example

  • For a scenario with a probability of success p=0.2:
    image
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