Due date: Thursday 2/12, 11:59pm
Poisson process
01
05
Link to originalPotholes on the highway
On a certain terrible stretch of highway, the appearance of potholes can be modeled by a Poisson process. Let the RV
denote the distance between successive potholes (measured in miles). The CDF of X is: (a) What is the mean number of potholes in a 2-mile stretch of the highway?
(b) What is the probability that there will be at least 2 potholes in a 2-mile stretch of the highway?
Solution
Solutions - 5160-05
(a)
, so the Poisson rate for a 2-mile stretch is , giving . The mean number of potholes in a 2-mile stretch is
.
(b)
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02
04
Link to originalSelling Christmas trees
An online company sells artificial Christmas trees. During the holiday season, the amount of time between sales,
, is an exponential random variable with an expected value of 2.5 hours. (a) Find the probability that the store will sell more than 2 trees in a 1-hour period of time.
(b) Find the probability the time between the sales of two trees will be between 4-5 hours.
Solution
Solutions - 5160-04
(a)
(b)
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Derived random variable
03
01
Link to originalConstants in PDF from expectation
Suppose
has PDF given by: Suppose
. Find the only possible values for and . Then find .
Solution
Solutions - 5150-01
(1) Recall formula for expectation of a continuous random variable:
(2) Use formula to find an equation relating
and :
(3) Integrate the PDF to get a second equation:
Since integrating a PDF should yield
:
(4) Solve system of equations for
and : Isolating
in the second equation yields . Plugging this into the first equation yields
. Solving for
yields , and thus .
(5) Compute variance:
Using
, first compute : So:
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04
02
Link to originalVariance: Direct integral formula
Suppose
has PDF given by: Find
directly using the integral formula.
Solution
Solutions - 5150-02
(1) Recall the integral formula for variance:
Use the fact that
.
(2) Compute
:
(3) Compute
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Review problems
05
05
Link to originalBinomial - Repeated coin flips
A coin is flipped 7 times and the sequence of results recorded as an outcome.
(a) How many possible outcomes have exactly 3 heads?
(b) How many possible outcomes have at least 3 heads?
Solution
Solutions - 5070-05
(a)
Out of
trials, we choose of them to be heads.
(b)
Out of 7 trials, we choose at least 3 of them to be heads.
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