01
Mean and variance of exponential
Show that
and for .
Solution
09
(1) State the PDF of an exponential distribution.
(2) Compute
using the integral formula.
(3) Compute
using the integral formula.
(4) Compute
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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.)
Solution
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.
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- Since both cars have the same remaining lifetime distribution, the probability that either car outlives the other is 0.5.
03
Wait time for 5 calls - two methods
Consider the Poisson process of phone calls coming to a call center at an average rate of 1 call every 6 minutes.
Let us model the wait time for 5 calls to come in. You may use Desmos or similar to perform the integration numerically.
(a) Method One: An arrival of ‘1-call’ comes in at an average rate of
calls per hour. So a Bundle of ‘5-calls’ comes in at an average rate of Bundles per hour. Use an exponential variable with to determine the probability that the wait time for a Bundle (of 5 calls) is at most . (b) Method Two: Use
calls per hour with an Erlang distribution at to determine the probability that the wait time for 5 calls is at most . (c) Compare the results of (a) and (b). Can you explain why they agree or disagree? Which is correct??
Solution
11
(a)
Compute probability the wait time for a Bundle is at most 1 hr.
Our bounds will be from 0 to 1 since we are only concerned about 1 hour.
(b)
State the Erlang distribution.
Compute desired probability.
(c)
State conclusions.
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- Clearly, the results disagree. This is because method 1 considers calls coming in at bundles at a time instead of considering 5 discrete calls. Method 2 is more accurate since it considers the rates of individual calls.
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.
Solution
15
(a)
(b)
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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?
Solution
14
(a)
(b)
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