01
Poisson satisfies
Show that a Poisson variable
satisfies the total probability rule for a CDF, namely that .
Solution
01
(1) State CDF of a Poisson distribution.
We know that
We know that
.
(2) Compute limit as
. Note that
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02
Expectation of Poisson
Derive the formula
for a Poisson variable .
Solution
02
(1) State PMF of a Poisson distribution.
(2) Find expectation.
We know that
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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?
Solution
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
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04
Silver dimes
Suppose 1 out of 500 dimes in circulation is made of silver. Consider a tub of dimes worth $40.
(a) Find a formula for the exact probability that this collection contains at least 2 silver dimes. (Do not use the complement event, you must perform a summation.) Can your calculator evaluate this formula?
(b) Estimate the probability in question using a Poisson approximation.
(This topic for HW only, not for tests.)
Solution
04
(a)
(1) Identify distribution.
Clearly, this scenario follows a binary distribution.
We have a
chance that the dime is made of silver. Since we have
40$ 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
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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
05
(a)
(1) Define random variable that is the Poisson approximation to
.
.
(2) Estimate error.
(b)
(1) Compute
using the binomial distribution. We have that
.
(2) Compute
using the Poisson distribution.
(3) Compute error.
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