AP Statistics Curriculum 2007 Limits Norm2Poisson

(Difference between revisions)
 Revision as of 04:57, 26 October 2009 (view source)IvoDinov (Talk | contribs) (ad)← Older edit Current revision as of 13:43, 18 September 2014 (view source)IvoDinov (Talk | contribs) (→Examples) (6 intermediate revisions not shown) Line 10: Line 10: ===Examples=== ===Examples=== Suppose cars arrive at a parking lot at a rate of 50 per hour.  Let’s assume that the process is a Poisson random variable with $\lambda=50$.  Compute the probability that in the next hour the number of cars that arrive at this parking lot will be between 54 and 62.  We can compute this as follows: Suppose cars arrive at a parking lot at a rate of 50 per hour.  Let’s assume that the process is a Poisson random variable with $\lambda=50$.  Compute the probability that in the next hour the number of cars that arrive at this parking lot will be between 54 and 62.  We can compute this as follows: - $P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617.$  The figure below from SOCR shows this probability. + $P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617.$  The figure below from the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Poisson Distribution] shows this probability.
[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]
[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]

Applications: Positron Emission Tomography

The physics of positron emission tomography (PET) provides evidence that the Poisson distribution model may be used to study the process of radioactive decay using positron emission. As tracer isotopes decay, they give off positively charged electrons which collide with negatively charged electrons the result of which (by the law of preservation of energy) is one or a pair of photons emitted at the annihilation point in space and detected by photo-multiplying tubes surrounding the imaged object (e.g., a human body part like the brain). The (large) number of arrivals at each detector is a Poisson process, which can be approximated by Normal distribution, as described above. This figure shows the schematics of the PET imaging technique.