AP Statistics Curriculum 2007 Limits Norm2Poisson - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

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Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov: /* Examples */

2014-09-18T13:54:52Z

‎Examples

IvoDinov: added see also and Applications: Positron Emission Tomography sections

2011-01-26T22:38:31Z

added see also and Applications: Positron Emission Tomography sections

Jenny: /* Examples */

2010-06-28T20:23:32Z

‎Examples

Jenny at 17:08, 28 June 2010

2010-06-28T17:08:22Z

IvoDinov: /* Examples */

2010-03-19T16:27:44Z

‎Examples

IvoDinov: /* Examples */

2010-03-19T16:27:05Z

‎Examples

IvoDinov: ad

2009-10-26T05:03:42Z

PriscillaChui: /* Normal Approximation to Poisson Distribution */

2008-03-01T22:20:54Z

‎Normal Approximation to Poisson Distribution

IvoDinov: /* Normal Approximation to Poisson Distribution */

2008-02-03T02:05:34Z

‎Normal Approximation to Poisson Distribution

← Older revision		Revision as of 18:01, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

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 <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]</center>
-* '''Note''': We observe that this distribution is bell-shaped.  We can use the normal distribution to approximate this probability.  Using <math> N(\mu=50, \sigma=\sqrt{50}=7.071) </math>, together with the continuity correction for better approximation we obtain <math> P(54 \le X \le 62)=0.2718 </math>, which is close to the exact that was found earlier.  The figure below shows this probability using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Normal Distribution Applet].
+* '''Note''': We observe that this distribution is bell-shaped.  We can use the normal distribution to approximate this probability.  Using <math> N(\mu=50, \sigma=\sqrt{50}=7.071) </math>, together with the continuity correction for better approximation we obtain $P(53.5<X<62.5| N(μ=50,σ=7.071))= 0.271759 $, which is close to the exact that was found earlier.  The figure below shows this probability using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Normal Distribution Applet].
 <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure12.jpg|600px]]</center>

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	<hr>		<hr>
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		+	===Applications: Positron Emission Tomography===
		+	The [http://en.wikipedia.org/wiki/Positron_emission_tomography 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 [http://en.wikipedia.org/wiki/Electron-positron_annihilation 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.
		+	<center>[[Image: SOCR_Activities_ExploreDistributions_Figure12.png\|600px]]</center>
		+
		+	=== See also===
		+	[[AP_Statistics_Curriculum_2007_Distrib_Poisson\| Poisson Distribution Section of the EBook]].

	===[[EBook_Problems_Limits_Norm2Poisson\|Problems]]===		===[[EBook_Problems_Limits_Norm2Poisson\|Problems]]===

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 <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]</center>
-* '''Note''': We observe that this distribution is bell-shaped.  We can use the normal distribution to approximate this probability.  Using <math> N(\mu=50, \sigma=\sqrt{50}=7.071) </math>, together with the continuity correction for better approximation we obtain <math> P(54 \le X \le 62)=0.2718 </math>, which is close to the exact that was found earlier.  The figure below shows this probability using hte [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Normal Distribution Applet].
+* '''Note''': We observe that this distribution is bell-shaped.  We can use the normal distribution to approximate this probability.  Using <math> N(\mu=50, \sigma=\sqrt{50}=7.071) </math>, together with the continuity correction for better approximation we obtain <math> P(54 \le X \le 62)=0.2718 </math>, which is close to the exact that was found earlier.  The figure below shows this probability using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Normal Distribution Applet].
 <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure12.jpg|600px]]</center>

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 ===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 <math> \lambda=50 </math>.  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:
-<math> P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617. </math>  The figure below from [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Poisson Distribution] shows this probability.
+<math> P(54 \le X \le 62) = \sum_{x=54}^{62} \frac{50^x e^{-50}}{x!}=0.2617. </math>  The figure below from the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Poisson Distribution] shows this probability.
 <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure11.jpg|600px]]</center>

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 <hr>
-===References===
+===[[EBook_Problems_Limits_Norm2Poisson|Problems]]===
 <hr>

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 === Normal Approximation to Poisson Distribution===
-The [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(<math> \lambda </math>) distribution]] can be approximated with [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal]] when <math> \lambda </math> is large.
+The [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(<math> \lambda </math>) Distribution]] can be approximated with [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal]] when <math> \lambda </math> is large.
-For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal(<math>\mu=\lambda, \sigma^2=\lambda</math>)]] 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.
+For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal(<math>\mu=\lambda, \sigma^2=\lambda</math>)]] 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.
 If <math>x_o</math> is a non-negative integer, <math>X\sim Poisson(\lambda)</math> and <math>U\sim Normal(\mu=\lambda, \sigma^2=\lambda</math>), then <math>P_X(X<x_o) = P_U(U<x_o+0.5)</math>.