Due date: Tuesday 9/30, 11:59pm
Poisson process
01
02
Expectation of Poisson
Derive the formula
for a Poisson variable . Link to originalSolution
02
(1) State PMF of a Poisson distribution.
(2) Find expectation.
We know that
. Link to original
02
04
Silver dimes
Suppose 1 out of 500 dimes in circulation is made of silver. Consider a tub of dimes worth $40.
(a) Find a formula for the exact probability that this collection contains at least 2 silver dimes. (Do not use the complement event, you must perform a summation.) Can your calculator evaluate this formula?
(b) Estimate the probability in question using a Poisson approximation.
(This topic for HW only, not for tests.)
Link to originalSolution
04
(a)
(1) Identify distribution.
Clearly, this scenario follows a binary distribution.
We have a
chance that the dime is made of silver. Since we have
40$ worth of dimes, there are 400 dimes. Thus,
.
(2) Find formula for probability.
(b)
(1) Find corresponding Poisson distribution.
Let
.
.
(2) Compute probability.
We have that
. Link to original
Function on a random variable
03
05
Expectation from CDF
The CDF of random variable X is given by:
Compute
. Link to originalSolution
16
(a)
(b)
Link to original
04
03
PDF of derived variable for
and Suppose the PDF of an RV is given by:
(a) Find
using the integral formula. (b) Find
, the PDF of (by calculating the CDF first). (c) Find
using . (d) Find
using results of (a) and (c). Link to originalSolution
08
(a)
Compute
.
(b)
(1) Find the CDF of
.
(2) Find the CDF of
. Since
is monotone increasing, .
(3) Find the PDF of
by differentiating.
(c)
Find
.
(d)
Compute
. Link to original
Continuous wait times
05
01
Mean and variance of exponential
Show that
and for . Link to originalSolution
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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06
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??
Link to originalSolution
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.
Review problems
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03
Rolling until a six
A fair die is rolled until a six comes up.
What are the odds that it takes at least 10 rolls? (Use a geometric random variable.)
Link to originalSolution
13
(1) Define random variables.
Let
. We wish to find
. For all
, , the first trials result in failure, and the trial is a success.
(2) Compute probability.
Note that the summation is simplified using the formula for a geometric series.
Link to original