AP Statistics Curriculum 2007 Distrib Poisson - Revision history

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

2020-03-03T19:37:31Z

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

Dinov: /* Example */

2014-09-25T13:04:06Z

‎Example

Dinov: /* Example */

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‎Example

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Dinov: /* Example */

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Dinov: /* Example */

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IvoDinov: /* Poisson Random Variables and Experiments */

2012-02-03T16:52:25Z

‎Poisson Random Variables and Experiments

IvoDinov: added see also section

2011-01-26T22:11:00Z

added see also section

Jenny at 17:02, 28 June 2010

2010-06-28T17:02:07Z

← Older revision		Revision as of 19:37, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Poisson}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Poisson}}

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 <center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig7.jpg|500px]]</center>
-* '''Note''': We observe that this distribution is bell-shaped.  We can also use the Normal distribution to approximate this probability. Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. Using <math>Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)</math>, together with the continuity correction for better approximation we obtain $P(43.5 \le X \le 52)=P\big (43.5 \le Y \le 52.5 | N(\mu=40, \sigma=6.325) \big )=0.265947$, which is close to the exact that was found earlier.  The figure below shows this probability.
+* '''Note''': We observe that this distribution is bell-shaped.  We can also use the Normal distribution to approximate this probability. Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. Using <math>Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)</math>, together with the continuity correction for better approximation we obtain $P(43.5 \le X \le 52)=P\big (43.5 \le Y \le 52.5 | N(\mu=40, \sigma=6.325) \big )=0.265947$, which is close to the exact probability ($0.255719$) that was found earlier.  The figure below shows this probability.
 <center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig8.jpg|500px]]</center>

@@ Line 27: / Line 27: @@
 <center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig7.jpg|500px]]</center>
-* '''Note''': We observe that this distribution is bell-shaped.  We can also use the Normal distribution to approximate this probability. Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. Using <math>Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)</math>, together with the continuity correction for better approximation we obtain $P(43.5 \le X \le 52)=P(43.5 \le Y \le 52.5 | N(\mu=40, \sigma=6.325)=0.265947$, which is close to the exact that was found earlier.  The figure below shows this probability.
+* '''Note''': We observe that this distribution is bell-shaped.  We can also use the Normal distribution to approximate this probability. Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. Using <math>Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)</math>, together with the continuity correction for better approximation we obtain $P(43.5 \le X \le 52)=P\big (43.5 \le Y \le 52.5 | N(\mu=40, \sigma=6.325) \big )=0.265947$, which is close to the exact that was found earlier.  The figure below shows this probability.
 <center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig8.jpg|500px]]</center>

@@ Line 27: / Line 27: @@
 <center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig7.jpg|500px]]</center>
-* '''Note''': We observe that this distribution is bell-shaped.  We can also use the Normal distribution to approximate this probability. Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. Using <math>Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)</math>, together with the continuity correction for better approximation we obtain $P(43.5 \le X \le 52)=P(43.5 \le Y \le 52.5 | N(\mu=40, \sigma=6.325)=265947$, which is close to the exact that was found earlier.  The figure below shows this probability.
+* '''Note''': We observe that this distribution is bell-shaped.  We can also use the Normal distribution to approximate this probability. Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. Using <math>Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)</math>, together with the continuity correction for better approximation we obtain $P(43.5 \le X \le 52)=P(43.5 \le Y \le 52.5 | N(\mu=40, \sigma=6.325)=0.265947$, which is close to the exact that was found earlier.  The figure below shows this probability.
 <center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig8.jpg|500px]]</center>

@@ Line 27: / Line 27: @@
 <center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig7.jpg|500px]]</center>
-* '''Note''': We observe that this distribution is bell-shaped.  We can also use the Normal distribution to approximate this probability.  Using <math>Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)</math>, together with the continuity correction for better approximation we obtain <math> P(44 \le X \le 52)=P(44 \le Y \le 52)=0.234655</math>, which is close to the exact that was found earlier.  The figure below shows this probability.
+* '''Note''': We observe that this distribution is bell-shaped.  We can also use the Normal distribution to approximate this probability. Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed. Using <math>Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)</math>, together with the continuity correction for better approximation we obtain $P(43.5 \le X \le 52)=P(43.5 \le Y \le 52.5 | N(\mu=40, \sigma=6.325)=265947$, which is close to the exact that was found earlier.  The figure below shows this probability.
 <center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig8.jpg|500px]]</center>

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 * Compute the probability that in a 1 hour timeframe, the number of observed shooting stars will be between 44 and 52! We can compute this as follows: Let <math>X\sim Poisson(\lambda)</math>.
 <math> P(44 \le X \le 52) = \sum_{x=44}^{52} \frac{40^x e^{-40}}{x!}=0.255719.</math>  The figure below shows this result using [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR distributions]:
-<center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig7.jpg]]</center>
+<center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig7.jpg|500px]]</center>
 * '''Note''': We observe that this distribution is bell-shaped.  We can also use the Normal distribution to approximate this probability.  Using <math>Y\sim N(\mu=40, \sigma=\sqrt{40}=6.325)</math>, together with the continuity correction for better approximation we obtain <math> P(44 \le X \le 52)=P(44 \le Y \le 52)=0.234655</math>, which is close to the exact that was found earlier.  The figure below shows this probability.
-<center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig8.jpg|600px]]</center>
+<center>[[Image:SOCR_EBook_Dinov_RV_Poisson_013008_Fig8.jpg|500px]]</center>
 * More about [[SOCR_EduMaterials_Activities_Explore_Distributions |approximation of Poisson by Normal distribution is available here]].

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 * Notes
-** The Poisson Distribution can be derived as a limiting case of the [[AP_Statistics_Curriculum_2007_Distrib_Binomial | Binomial Distribution]].
+** The [[AP_Statistics_Curriculum_2007_Distrib_Poisson#Poisson_as_a_Limiting_Case_of_Binomial_Distribution|Poisson Distribution can be derived as a limiting case]] of the [[AP_Statistics_Curriculum_2007_Distrib_Binomial | Binomial Distribution]].
 ** The Poisson Distribution can be applied to systems with a large number of possible events, each of which is rare. A classic example is the nuclear decay of atoms and [http://en.wikipedia.org/wiki/Positron_emission_tomography Positron Emission Tomography Imaging].

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 :<math>\lim_{n\rightarrow\infty} np_n = \lambda,</math>
 the sequence converges  (in distribution) to a Poisson random variable with mean λ.
 ===[[EBook_Problems_Distrib_Poisson|Problems]]===