Bernoulli and Binomial Distributions
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Questions and Answers

What is the probability mass function (p.m.f.) of a Bernoulli distribution with parameter (p)?

  • \(p^x(1-p)^x\)
  • \(p^{1-x}(1-p)^x\)
  • \(p(1-p)^x\)
  • \(p^x(1-p)^{1-x})\) (correct)

    • What is the variance of a Bernoulli distribution with parameter (p)?

  • \(1 - p\)
  • \(p(1-p)^2\)
  • \(p(1 - p)\) (correct)
  • \(p^2(1-p)\)

    • A coin is flipped 10 times. What is the probability of getting exactly 6 heads, assuming the coin is fair?

  • \(10 \choose 6 / 2^{10}\) (correct)
  • \(1/2^{10}\)
  • \(10! / (6! imes 4!)\)
  • \(6/10\)

    • What are the parameters of a binomial distribution?

    <p>The probability of success and the number of trials (B)</p> Signup and view all the answers

    What is the mean of a binomial distribution with (n) trials and probability of success (p)?

    <p>(np) (C)</p> Signup and view all the answers

    A fair die is rolled 5 times. What is the variance of the number of times a 6 appears?

    <p>(25/36) (C)</p> Signup and view all the answers

    What is the range of a binomial distribution with (n) trials?

    <p>(0, 1, 2, ..., n) (A)</p> Signup and view all the answers

    Which of the following is NOT a feature of a binomial distribution?

    <p>The outcomes must be continuous. (C)</p> Signup and view all the answers

    Flashcards

    Bernoulli Distribution

    A distribution for a single trial with two outcomes: success or failure.

    Probability Mass Function (p.m.f.)

    The formula: p(x) = p^x(1-p)^(1-x), for x=0,1 in Bernoulli distribution.

    Variance of Bernoulli

    The measure of spread in a Bernoulli distribution: Variance = p(1-p).

    Binomial Distribution

    A distribution based on the number of successes in n independent Bernoulli trials.

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    Mean of Binomial Distribution

    The average number of successes: μ = np, where n is trials and p is success probability.

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    Variance of Binomial Distribution

    The spread in a binomial distribution: σ² = np(1-p).

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    Standard Deviation of Binomial Distribution

    The square root of variance: σ = √(np(1-p)).

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    Range of Binomial Distribution

    The possible outcomes: {0, 1, 2, …, n} where n is the number of trials.

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    Study Notes

    Bernoulli Distribution

    • A Bernoulli distribution models a single trial with two possible outcomes (success or failure).
    • The probability mass function (PMF) is p(x) = px(1-p)1-x, where x = 0 or 1.
    • Parameter: p (probability of success)
    • Mean = p
    • Variance = p(1-p)

    Binomial Distribution

    • A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials.
    • Parameter: n (number of trials), p (probability of success in a single trial)
    • Probability mass function (PMF): P(X=k) = nCk * pk * (1-p)(n-k)
    • Range: 0 ≤ X ≤ n , where X is the number of successes
    • Mean = np
    • Variance = np(1-p)
    • Standard Deviation = √(np(1-p))

    Relationships

    • The mean of a binomial distribution is equal to the product of the number of trials and the probability of success in a single trial.

    • The variance of a binomial distribution is equal to the product of the number of trials, the probability of success, and the probability of failure in a single trial.

    • A recurrence relation for successive probabilities in a binomial distribution involves the probabilities of different numbers of successes in n trials, considering the probability of success (p).

    • A recurrence relation for successive frequencies in a binomial distribution involves the frequencies of different numbers of successes in n trials, considering the probability of success (p).

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    Description

    This quiz covers essential concepts related to Bernoulli and Binomial distributions, including their probability mass functions, parameters, means, variances, and relationships. Test your understanding of how these distributions are related and their applications in probability theory.

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