01
Poisson satisfies
Show that a Poisson variable
satisfies the total probability rule for a CDF, namely that .
Solution
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 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?
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.)
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 probability
. (This topic for HW only, not for tests.)
Solution