Central Limit Theorem in Probability Theory
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Questions and Answers

What does the central limit theorem (CLT) state?

  • The distribution of a normalized sample mean converges to a standard normal distribution. (correct)
  • The distribution of a normalized sample mean converges to an exponential distribution.
  • The distribution of a normalized sample mean converges to a standard uniform distribution.
  • The distribution of a normalized sample mean converges to a Poisson distribution.

    • Why is the central limit theorem (CLT) considered a key concept in probability theory?

  • It is only applicable in certain specific conditions.
  • It implies that probabilistic methods for normal distributions can be extended to other types of distributions. (correct)
  • It contradicts other fundamental theorems in probability theory.
  • It only applies to normal distributions.

    • When was the modern general form of the central limit theorem (CLT) precisely stated?

  • 2000
  • 1945
  • 1811
  • 1920 (correct)

    • Which type of variables need to be normally distributed for the central limit theorem (CLT) to hold?

    <p>Neither original variables nor sample mean</p> Signup and view all the answers

    What is the main implication of the central limit theorem (CLT) on statistical methods?

    <p>Statistical methods for normal distributions can be applied to many other types of distributions.</p> Signup and view all the answers

    Which version of the central limit theorem (CLT) serves as a bridge between classical and modern probability theory?

    <p>Modern general form from 1920</p> Signup and view all the answers

    What does the central limit theorem state about the probability distribution of observed averages?

    <p>It approximates a normal distribution</p> Signup and view all the answers

    In the classical central limit theorem, what does the difference between the sample average and its limit approximate?

    <p>Normal distribution with mean 0</p> Signup and view all the answers

    What type of random variables are required in the common form of the central limit theorem?

    <p>Independent and identically distributed (i.i.d.)</p> Signup and view all the answers

    What is the factor that the difference between the sample average and its limit is multiplied by in the classical central limit theorem?

    <p>$rac{1}{n}$</p> Signup and view all the answers

    What does the law of large numbers state about the sample average as n approaches infinity?

    <p>It approaches the expected value</p> Signup and view all the answers

    Which theorem states that the normal distribution can be used as an approximation to the binomial distribution?

    <p>De Moivre–Laplace Theorem</p> Signup and view all the answers

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