Due date: Friday 9/26, 9:00am
Poisson process
01
03
Application of Poisson: meteor shower
The UVA astronomy club is watching a meteor shower. Meteors appear at an average rate of
per hour. (a) Write a short explanation to justify the use of a Poisson distribution to model the appearance of meteors. Why should appearances be Poisson distributed?
(b) What is the probability that the club sees more than 2 meteors in a single hour?
(c) Suppose we learn that over a four hour evening, 13 meteors were spotted. What is the probability that none of them happened in the first hour?
Link to originalSolution
03
(a)
Write explanation.
- You use Poisson distribution if events occur randomly and you know the mean number of events that occur within a given interval of time.
- In addition, Poisson distributions are advantageous when describing rare events. Since meteors are a rare occurrence, it makes sense to use a Poisson distribution.
(b)
Compute probability.
Since
, it’s easier to compute the latter.
(c)
Compute probability.
We know that there are 13 meteors in 4 hours, so we see an average of
meteors per hour. Let We wish to find the probability
. Link to original
02
05
Potholes 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?
Link to originalSolution
14
(a)
(b)
Link to original
Function on a random variable
03
01
Constants in PDF from expectation
Suppose
has PDF given by: Suppose
. Find the only possible values for and . Then find . Link to originalSolution
06
(1) Recall formula for expectation of a continuous random variable.
(2) Use formula to find an equation relating
and .
(3) Recall that integrating a PDF should yield
. Integrate the PDF and find a second equation relating and .
(4) Solve system of equations for
and . Isolating
in the second equation yields . Plugging this expression into the first equation yields
. Solving for
yields 2.4, and thus .
(5) Compute variance using the formula
Compute
. Link to original
04
02
Variance: Direct integral formula
Suppose
has PDF given by: Find
directly using the integral formula. Link to originalSolution
07
(1) Recall the integral formula for variance.
Use the fact that
(2) Compute
.
(3) Compute
Link to original
Continuous wait times
05
02
Vehicle lifetimes
Suppose that vehicle lifetimes follow an exponential distribution with an expected lifetime of 10 years.
Suppose you have one car that is 5 years old, and one that is 15 years old, at the present moment.
What is the probability that the first car outlives the second? (I.e. that the second breaks at an earlier time than the first breaks, both starting now.)
Link to originalSolution
10
Recall the memoryless property of exponential distributions.
- Elapsed time has no effect on future events.
- Therefore, the fact that one car is older than the other has no effect on the remaining lifetimes.
Derive conclusions.
Link to original
- Since both cars have the same remaining lifetime distribution, the probability that either car outlives the other is 0.5.
06
04
Selling 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.
Link to originalSolution
15
(a)
(b)
Link to original
Review problems
07
05
Binomial - 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?
Link to originalSolution
12
(a)
Out of
trials, we choose of them to be heads. Thus,
(b)
Out of 7 trials, we choose at least 3 of them to be heads.
Using summation notation, we get
Link to original