Math 337 - Practice Problems #1
Instructions: These problems are practice for the test problems. You may assume that you will
be provided with:
• The formulas for the probability distribution functions we have covered in class, and
• If long sums are required (and unavoidable using standard mathematical identities; see
Question 4) for a cumulative distribution (e.g. more than 4 terms), you will be given the
table columns for the relevant cumulative distributions
1. In a sequence of 7 coin flips, find:
(a) The probability distribution (formula) for the number of tails that appears.
(b) The probability of exactly three tails.
(c) The probability of exactly five heads.
(d) The probability of three or fewer tails.
2. In a sequence of 8 dice rolls, where you ‘win’ on a roll of a 5 or a 6, find:
(a) The probability distribution over the number of wins (formula).
(b) The probability of exactly four wins.
(c) The probability of up to and including two wins.
(d) Find the expected number of wins.
3. An emergency room in Ottawa averages 20 head injury cases per week (some major, some
minor; we won’t distinguish between them).
(a) Find a formula for the probability distribution for the number of head injuries.
(b) Find the probability of a light week, defined as 15 or fewer head injuries coming in. You
may want to use a table for this question.
(c) Find the probability of a bad week, defined as 25 or more head injuries.
(d) Use a calculation (not the table) to find the probability of exactly 20 head injuries
arriving at the ER. Confirm your answer by referring to the table of cumulative values.
(e) If you consider the wider geographic region, which has an average of 240 injuries per
week, and try to compute the probability of exactly 200 injuries, you’ll reach past the
limit of what your calculator (and standard software) can handle (Give it a try!) What
approaches might be helpful in overcoming this problem?
4. In successive rolls of a fair die, let X be the number of rolls until the first 6 appears.
Determine the probability distribution and the cumulative distribution function of X.
5. In a call center, the time to deal with a customer are distributed exponentially, with a mean
of 2.3 minutes.
1
(a) Find the probability that a call takes less than 2 minutes.
(b) Find the probability that a call takes more than 5 minutes.
6. We know quick rules of thumb for the normal distribution (e.g. “95% of the probability lies
within two standard deviations of the mean”). We may find it helpful to have similar rules
for the exponential distribution.
Consider an
probabilities
(a) P X <
(b) P X <
exponentially distributed inter-arrival time, with mean λ minutes. Find the
that
λ
?
4
λ
?
2
(c) P (X < λ)?
(d) P (X > 2λ)?
(e) P (X > 4λ)?
(f) Try to summarize each result in a sentence, or put them in a chart.
7. The number of days that elapse between the beginning of a calendar and the moment a highrisk driver is involved in an accident is exponentially distributed. An insurance company
expects that 30% of high-risk drivers will be involved in an accident during the first 50 days
of a calendar year. What portion of high-risk drivers are expected to be involved in an
accident during the first 80 days of a calendar year?
8. The memory-less nature of the exponential function can be written mathematically as
P (X > a) = P (X > t + a|X > t)
(1)
In words, the probability that the time will be greater than a, relative to time zero, is the
same as the probability that the time will be greater than t + a, given that we’ve already
lasted to time t). Use the parameter θ for the average time.
T
P (A B)
to rewrite
(a) Use the fact that conditional probabilities can be written P (A|B) =
P (B)
the RHS of Equation (1) as a fraction of two probabilities.
(b) Look at the numerator: there are two conditions in the intersection, but they can be
replaced by a single condition. Simplify the numerator accordingly.
(c) As a useful side formula, compute the probability P (X > k) for the exponential distribution.
(d) Using the above formula when necessary, rewrite both probabilities from part 8b using
the exponential distribution, simplify, and rewrite as a probability to complete the proof
of memoryless property.
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9. The time it takes for a student to finish an aptitude test (in hours) has a density function
of the form
(
c(y − 1)(2 − y) 1 < y < 2
f (y) =
0
otherwise
(a) Determine the constant c that makes f (y) a proper PDF.
(b) What is the cumulative distribution function of the time it takes for a randomly selected
student to finish the aptitude test?
(c) What is the probability that a student will finish the aptitude test in less than 75
minutes?
(d) What is the probability that a student will finish the aptitude test between 1 1/2 and
2 hours?
10. Suppose there are 4 riflemen equipped with inferior weapons, in the sense that the time
to weapon failure is exponentially distributed with a mean of 3 minutes. Suppose the first
rifleman fires until his/her weapon fails, then the second rifleman starts firing. When the
second rifleman’s weapon fails, the third rifleman starts firing, etc. What is the probability
that the riflemen will fire continuously for 10 minutes or more? (Warning: this will involve
integration by parts applied several times)
11. Assume that the probability of rain tomorrow is 0.5 if it is raining today, and assume that the
probability of its being clear (no rain) tomorrow is 0.9 if it is clear today. Also assume that
these probabilities do not change if additional information is provided about the weather
prior to today.
(a) Explain why the stated assumptions imply that the Markovian property holds for the
evolution of the weather.
(b) Formulate the evolution of the weather as a Markov chain by defining its states.
(c) Determine the one-step transition matrix of the Markov chain.
(d) Find the two-step transition matrix.
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