The symbol
Poisson process
01
Poisson satisfies
Show that a Poisson variable
satisfies the total probability rule for a CDF, namely that .
02
Expectation of Poisson
Derive the formula
for a Poisson variable .
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 that over a four hour evening, 13 meteors were spotted. What is the probability that none of them happened in the first hour?
04
Silver dimes
Suppose 1 out of 350 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. Can your calculator evaluate this formula?
(b) Estimate the probability in question using a Poisson approximation.
(This topic for HW only, not for tests.)
05
Application of Poisson approximation of binomial
Let
and consider the Poisson approximation to . (a) Estimate the possible error of the approximation (for an arbitrary probability).
(b) Compute the exact error of the approximation for the specific value
. (This topic for HW only, not for tests.)
Function on a random variable
06
Constants in PDF from expectation
Suppose
has PDF given by: Suppose
. Find the only possible values for and . Then find .
07
Variance: Direct integral formula
Suppose
has PDF given by: Find
using the integral formula.
08
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).
Continuous wait times
09
Mean and variance of exponential
Show that
and for .
10
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.)
11
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 wait 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??