Table of Contents

  • Theorem 1.15: Binomial Distribution
      - Definition of Independent Bernoulli Random Variables
      - Definition of X and its Distribution
  • The Geometric Distribution (Pascal Distribution)
      - Definition 1.18

Theorem 1.15: Binomial Distribution

  • Definition: Let ( Y_1, Y_2, …, Y_n ) be independent Bernoulli random variables, each with parameter ( p ).
  • Bernoulli Random Variable: A Bernoulli random variable takes the value 1 with probability ( p ) (success) and the value 0 with probability ( 1-p ) (failure).
  • Sum of Bernoulli Variables: If ( X = Y_1 + Y_2 + … + Y_n ), then ( X ) counts the number of successes in ( n ) independent Bernoulli trials.
  • Distribution: ( X ) has the binomial distribution with parameters ( n ) (number of trials) and ( p ) (probability of success).
      - Binomial Distribution: Denoted as ( X \sim B(n, p) ). The probability mass function is given by:

    P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
      - Where:
        - ( k ) = number of successes (where ( k = 0, 1, 2, …, n ))
        - ( inom{n}{k} ) = binomial coefficient defined as ( rac{n!}{k!(n-k)!} )

The Geometric Distribution (Pascal Distribution)

  • Definition 1.18: The geometric distribution models the number of trials until the first success in a series of Bernoulli trials.

  • Characteristics:
      - Memoryless property: The probability of success in the future does not depend on past trials.
      - If ( X ) follows a geometric distribution with parameter ( p ), denoted as ( X \sim Geometric(p) ), then the probability mass function is:
    P(X=k)=(1−p)k−1pP(X = k) = (1-p)^{k-1} p
      - Where:
        - ( k ) = number of trials until the first success (where ( k = 1, 2, 3, … ))

  • Relation to Other Distributions: The geometric distribution can be seen as a special case of the negative binomial distribution, where one success is required.

  • Applications: Commonly used in reliability testing and survival analysis, the geometric distribution helps in understanding processes that involve waiting times until the first occurrence of an event.