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FRM Part I · FRM Exam Part I · Common Univariate Random Variables

Which statement about the relationship between the binomial and Poisson distributions is correct?

A binomial with large n and small p is well approximated by a Poisson distribution with lambda equal to np. In this limit the variance np(1-p) approaches np, matching the Poisson property that mean equals variance.

  1. AA binomial with large n and small p is approximated by a Poisson with lambda = npCorrect
  2. BA Poisson distribution has a variance equal to the square of its mean
  3. CA binomial distribution always has mean equal to variance
  4. DA Poisson with lambda = np is a good approximation of a binomial when p is near 0.5 and n is small

Explanation

As n grows large and p becomes small with np fixed, the binomial converges to a Poisson with lambda = np. A Poisson has variance equal to its mean, not its square. A binomial has variance np(1-p), which is less than the mean.

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