Master the Negative Binomial and Geometric Distributions in this Quiz!
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Questions and Answers

What is the sample space for the Negative Binomial Distribution?

  • S = {(w1 , w2 ,. , wn ) : n ∈ {r, r + 1,.}, wn = s, wi ∈ {s, f }, i = 1, 2,. , n − 1; r of w1 , w2 ,. , wn−1 are s and remaining n − r of w1 , w2 ,. , wn−1 are f }
  • S = {(w1 , w2 ,. , wn ) : n ∈ {r, r + 1,.}, wn = s, wi ∈ {s, f }, i = 1, 2,. , n; r − 1 of w1 , w2 ,. , wn−1 are s and remaining n − r of w1 , w2 ,. , wn−1 are f }
  • S = {(w1 , w2 ,. , wn ) : n ∈ {r, r + 1,.}, wn = s, wi ∈ {s, f }, i = 1, 2,. , n; r of w1 , w2 ,. , wn are s and remaining n − r of w1 , w2 ,. , wn are f }
  • S = {(w1 , w2 ,. , wn ) : n ∈ {r, r + 1,.}, wn = s, wi ∈ {s, f }, i = 1, 2,. , n − 1; r − 1 of w1 , w2 ,. , wn−1 are s and remaining n − r of w1 , w2 ,. , wn−1 are f } (correct)
  • Which of the following best describes the Negative Binomial Distribution?

  • A distribution that models the number of failures before the r-th success in a sequence of independent Bernoulli trials (correct)
  • A distribution that models the number of successes in a fixed number of independent Bernoulli trials
  • A distribution that models the number of successes before the r-th failure in a sequence of independent Bernoulli trials
  • A distribution that models the number of failures in a fixed number of independent Bernoulli trials
  • What does the random variable X represent in the Negative Binomial Distribution?

  • The number of failures in a fixed number of independent Bernoulli trials
  • The number of failures before the r-th success (correct)
  • The number of successes before the r-th failure
  • The number of successes in a fixed number of independent Bernoulli trials
  • What is the probability of the event {X = x} for x ∈ {0, 1, 2, · · · } in the Negative Binomial Distribution?

    <p>P ({X = x}) = 0</p> Signup and view all the answers

    What does the probability p represent in the Negative Binomial Distribution?

    <p>The probability of success in each trial</p> Signup and view all the answers

    Study Notes

    Sample Space of Negative Binomial Distribution

    • The sample space consists of non-negative integers: {0, 1, 2, ...} representing the number of failures before achieving a specified number of successes.

    Description of Negative Binomial Distribution

    • The Negative Binomial Distribution models the number of failures in a sequence of Bernoulli trials before a predetermined number of successes occurs.
    • It is a generalization of the geometric distribution, which focuses on the number of trials until the first success.

    Random Variable X in Negative Binomial Distribution

    • The random variable X represents the total number of failures encountered before achieving a specified number of successful trials (r).

    Probability of Event {X = x}

    • The probability mass function for the Negative Binomial Distribution is given by:
      • P(X = x) = C(x + r - 1, r - 1) * p^r * (1 - p)^x
      • Where C indicates the binomial coefficient, p is the probability of success on each trial, and x is the number of failures.

    Probability p in Negative Binomial Distribution

    • The probability p denotes the likelihood of success in each individual Bernoulli trial.
    • It is a key parameter and influences the shape of the distribution.

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    Description

    Test your understanding of the Negative Binomial and Geometric distributions with this quiz. Explore the concepts of success probability and the number of trials needed for a certain number of successes.

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