AP Statistics Curriculum 2007 Gamma - Revision history

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

2020-03-03T19:22:19Z

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

IvoDinov: /* Normal Approximation to Gamma distribution */

2012-06-23T17:40:30Z

‎Normal Approximation to Gamma distribution

IvoDinov: /* Normal Approximation to Gamma distribution */

2012-06-23T17:32:44Z

‎Normal Approximation to Gamma distribution

IvoDinov: /* Normal Approximation to Gamma distribution */

2012-06-23T17:32:24Z

‎Normal Approximation to Gamma distribution

IvoDinov: /* Normal Approximation to Gamma distribution */

2012-06-23T17:31:03Z

‎Normal Approximation to Gamma distribution

IvoDinov: /* Normal Approximation to Gamma distribution */

2012-06-23T17:30:42Z

‎Normal Approximation to Gamma distribution

IvoDinov: /* Normal Approximation to Gamma distribution */

2012-06-23T17:29:13Z

‎Normal Approximation to Gamma distribution

IvoDinov: /* Normal Approximation to Gamma distribution */

2012-06-23T17:28:05Z

‎Normal Approximation to Gamma distribution

IvoDinov: /* Normal Approximation to Gamma distribution */

2012-06-23T17:27:32Z

‎Normal Approximation to Gamma distribution

IvoDinov: /* Normal Approximation to Gamma distribution */

2012-06-23T17:14:24Z

‎Normal Approximation to Gamma distribution

← Older revision		Revision as of 19:22, 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/AP_Statistics_Curriculum_2007_Gamma}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php/AP_Statistics_Curriculum_2007_Gamma}}

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 ===Normal Approximation to Gamma distribution===
-Note that if \( \{X_1,X_2,X_3,\cdots \}\) is a sequence of independent [[AP_Statistics_Curriculum_2007_Exponential|Exponential(b) random variables]] then \(Y_k = \sum_{i=1}^k{X_i} \) is a [http://www.math.uah.edu/stat/special/Gamma.html random variable with gamma distribution with some shape parameter], k (positive integer) and scale parameter b. By the [[AP_Statistics_Curriculum_2007_Limits_CLT|central limit theorem]], if k is large, then gamma distribution can be approximated by the normal distribution with mean \(\mu=kb\) and variance \(\sigma^2 =kb^2\). That is, the distribution of the variable \(Z_k={{Y_k-kb}\over{\sqrt{k}b}}\) tends to the standard normal distribution as <math>k\longrightarrow \infty</math>.
+Note that if \( \{X_1,X_2,X_3,\cdots \}\) is a sequence of independent [[AP_Statistics_Curriculum_2007_Exponential|Exponential(b) random variables]] then \(Y_k = \sum_{i=1}^k{X_i} \) is a [http://www.math.uah.edu/stat/special/Gamma.html random variable with gamma distribution] with the following shape parameter, '''k''' (positive integer indicating the number of exponential variable in the sum) and scale parameter '''b''' (which is the exponential parameter). By the [[AP_Statistics_Curriculum_2007_Limits_CLT|central limit theorem]], if k is large, then gamma distribution can be approximated by the normal distribution with mean \(\mu=kb\) and variance \(\sigma^2 =kb^2\). That is, the distribution of the variable \(Z_k={{Y_k-kb}\over{\sqrt{k}b}}\) tends to the standard normal distribution as <math>k\longrightarrow \infty</math>.
 For the example above, \(\Gamma(k=4, \theta=2)\), the [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html SOCR Normal Distribution Calculator] can be used to obtain an estimate of the area of interest as shown on the image below.

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 {| class="wikitable" style="text-align:center; width:75%" border="1"
 |-
-! SUmmary|| [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html \(\Gamma(k=4, \theta=2)\) ] || [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html \(Normal(\mu=8, \sigma^2=4)\) ]
+! Summary|| [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html \(\Gamma(k=4, \theta=2)\) ] || [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html \(Normal(\mu=8, \sigma^2=4)\) ]
 |-
 | Mean||8.000000||8.0

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 {| class="wikitable" style="text-align:center; width:75%" border="1"
 |-
-! Probability|| [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html \(\Gamma(k=4, \theta=2)\) ] || [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html \(Normal(\mu=8, \sigma^2=4)\) ]
+! SUmmary|| [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html \(\Gamma(k=4, \theta=2)\) ] || [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html \(Normal(\mu=8, \sigma^2=4)\) ]
 |-
 | Mean||8.000000||8.0
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 |-
 | Max Density|| 0.112021||0.099736
 |-
 | <2|| 0.018988|| 0.066807

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 <center>[[Image:EBook_Gamma_Fig2.png|500px]]</center>
-The probabilities of the [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html real Gamma] and [[http://socr.ucla.edu/htmls/dist/Normal_Distribution.html approximate Normal]] distributions (on the range [2:4]) are not identical but are sufficiently close.
+The probabilities of the [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html real Gamma] and [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html approximate Normal] distributions (on the range [2:4]) are not identical but are sufficiently close.
 <center>

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 For the example above, \(\Gamma(k=4, \theta=2)\), the [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html SOCR Normal Distribution Calculator] can be used to obtain an estimate of the area of interest as shown on the image below.
-<center>[[Image:EBook_Gamma_Fig2.jpg|500px]]</center>
+<center>[[Image:EBook_Gamma_Fig2.png|500px]]</center>
 The probabilities of the ''real Gamma'' and ''approximate Normal'' distributions (on the range [2:4]) are not identical but are sufficiently close.

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 |-
 | >4|| 0.857123||0.841345
-\}
+|}
 </center>