AP Statistics Curriculum 2007 Limits Norm2Poisson

(Difference between revisions)
 Revision as of 18:56, 14 June 2007 (view source)IvoDinov (Talk | contribs)← Older edit Current revision as of 13:43, 18 September 2014 (view source)IvoDinov (Talk | contribs) (→Examples) (12 intermediate revisions not shown) Line 2: Line 2: === Normal Approximation to Poisson Distribution=== === Normal Approximation to Poisson Distribution=== - Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)! + The [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson( \lambda [/itex]) Distribution]] can be approximated with [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal]] when $\lambda is large. - [[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]] + - ===Approach=== + For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal([itex]\mu=\lambda, \sigma^2=\lambda$)]] Distribution is an excellent approximation to the [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(λ)]] Distribution.  If λ is greater than about 10, then the Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. - Models & strategies for solving the problem, data understanding & inference. + - * TBD + If $x_o$ is a non-negative integer, $X\sim Poisson(\lambda)$ and $U\sim Normal(\mu=\lambda, \sigma^2=\lambda$), then $P_X(X. Compute the probability that in the next hour the number of cars that arrive at this parking lot will be between 54 and 62. We can compute this as follows: + [itex] P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617.$  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]]

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.