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1. Identify each of the
following random variables as discrete or continuous. You do not need to
write the questions for #1, but you do
need to write the word “discrete” or “continuous” each time in order to
practice your spelling.
(a) age (to the nearest year) of a randomly selected citizen of Maine
(b) diameter of a cylindrical table leg turned on a lathe
(c) snowfall (to the nearest quarter inch) at a randomly selected location on
earth on a randomly selected day
(d) weight of a randomly chosen student
2. Mr. Hansen’s brother is a consistent 90% free-throw shooter. Mr. Hansen’s
brother is unflappable, and this probability never changes. Let X be the number of free throws made in
12 tries.
(a) Explain why X
is a binomial random variable. Be complete.
(b) List the sample space of possible outcomes for X, along with their associated probabilities.
(c) The set of all possible outcomes for a random variable, along with their
associated probabilities (or probability densities, in the case of a
continuous r.v.) is called a
____________________________ (hint:
starts with the letter D) and is usually depicted by means of a
__________________ __________________ __________________ (hint: letters R, F, H) or a
__________________ curve. We use the former depiction for __________________
random variables, the latter depiction for __________________ random
variables.
(d) Compute the probability that fewer than 8 free throws are made in 12
tries.
(e) Compute the probability that exactly 9 or 10 free throws are made in 12
tries.
3. There are about 600,000 people in Washington, DC, of whom approximately 2%
are HIV-positive. If you shake the hands of randomly chosen DC residents, one
by one, let Y denote the number of
hands you need to shake in order to encounter an HIV-positive individual.
(a) Explain why Y
is not a geometric random variable.
(b) Is it acceptable to treat Y as
if it were geometric? Why or why not?
(c) Compute the mean of Y.
(e) Compute the probability that the first HIV-positive person whose hand you
shake is person 35 or later.
(f) Compute the probability that the first HIV-positive person whose hand you
shake is strictly after person 15
but strictly before person 79.
4. Mr. Hansen’s latest SAT alternative test (the HAT,
or Hansen Aptitude Test) has a mean of 540 and a s.d.
of 110. Scores are approximately normally distributed. Compute
(a) the 60th percentile of the HAT
(b) the proportion of test takers who score between 580 and 640
(c) the HAT score that corresponds to an SAT score of 750, given that the SAT
is approximately N(500,100)
(d) the cutoff HAT scores that capture the central 72% of the distribution.
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