Time: 20 minutes. This quiz is open book, open notes. You must provide reasons for your answers.
1. Estimate the probability that a Binomial(144, 0.25) variable will have an outcome between 30 and 42. Justify your approximation method.
The figure shows PDFs for four distributions. Number 1
is symmetric, number 2 has the lowest peak, number 3 is discontinuous, and
number 4 has many modes. Distributions #1 and #2 have infinite support,
but almost all their probability is concentrated within the region shown here
and their tails continue to decrease very rapidly outside the region shown. #3
and #4 have finite support contained within the region shown here.
2. Which of these distributions have positive third central moment? Negative? Zero?
3. Which distribution has largest variance?
4. Of these four distributions, which one(s) will have the smallest 7th central moment?
5. (Extra credit) Estimate the probability that a Binomial(144, 0.25) variable will equal its mean. Justify your estimate.
Scoring: The passing score is 90.
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