SOCR EduMaterials Activities Binomial PGF - Revision history

WikiSysop: /* See also */

2014-03-05T14:55:25Z

‎See also

WikiSysop: /* This is an activity to explore the Probability Generating Functions for the Bernoulli, Binomial, Geometric and Negative-Binomial Distributions. */

2014-03-05T14:54:57Z

‎This is an activity to explore the Probability Generating Functions for the Bernoulli, Binomial, Geometric and Negative-Binomial Distributions.

Rgidwani at 18:36, 9 January 2008

2008-01-09T18:36:11Z

Rgidwani at 07:03, 9 January 2008

2008-01-09T07:03:50Z

IvoDinov at 06:31, 9 January 2008

2008-01-09T06:31:40Z

IvoDinov at 06:30, 9 January 2008

2008-01-09T06:30:45Z

Rgidwani at 03:39, 9 January 2008

2008-01-09T03:39:21Z

Rgidwani: New page: == This is an activity to explore the Probability Generating Functions for the Bernoulli, Binomial, Geometric and Negative-Binomial Distributions.== * '''Description''': You can access t...

2008-01-09T03:37:25Z

New page: == This is an activity to explore the Probability Generating Functions for the Bernoulli, Binomial, Geometric and Negative-Binomial Distributions.== * '''Description''': You can access t...

New page

== This is an activity to explore the Probability Generating Functions for the Bernoulli, Binomial, Geometric and Negative-Binomial Distributions.==

* '''Description''': You can access the applets for the above distributions at http://www.socr.ucla.edu/htmls/SOCR_DistributionFunctors.html .

* '''Exercise 1:''' Use SOCR to graph the PGF's and print the following distributions and answer the questions below. Also, comment on the shape of each one of these distributions:
**a.<math> X \sim Bernoulli(0.5) </math>
**b.<math> X \sim Binomial(1,0.5) </math>
**c.<math> X \sim Geometric(0.5) </math>
**d.<math> X \sim NegativeBinomial(1, 0.5) </math>

Below you can see a snapshot of the PGF of the distribution of <math> X \sim Bernoulli(0.8) </math>
<center>[[Image:BernoulliPGF.jpg|600px]]</center>

Do you notice any similarities between the graphs of these PGF's between any of these distributions?

* '''Exercise 2:''' Use SOCR to graph and print the PGF of the distribution of a geometric random variable with <math> p=0.2, p=0.7 </math>. What is the shape of this function? What happens when <math> p </math> is large? What happens when <math> p </math> is small?

* '''Exercise 3:''' You learned in class about the properties of PGF's If <math> X_1, ...X_n</math> are iid. and <math>Y = \sum_{i=1}^n X_i. </math> then <math>P_{y}(t) = {[P_{X_1}(t)]}^n</math>.
**a. Show that the PGF of the sum of <math>n</math> independent Bernoulli Trials with success probability <math> p </math> is the same as the MGF of the Binomial Distribution using the corollary above.
**b. Show that the PGF of the sum of <math>n</math> independent Geometric Random Variables with success probability <math> p </math> is the same as the MGF of the Negative-Binomial Distribution using the corollary above.
**c. How does this relate to Exercise 1? Does having the same MGF mean they are distributed the same?

@@ Line 27: / Line 27: @@
 <hr>
-* SOCR Home page: http://www.socr.ucla.edu
+* SOCR Home page: http://www.socr.umich.edu
-{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_Binomial_PGF}}
+{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=SOCR_EduMaterials_Activities_Binomial_PGF}}

← Older revision		Revision as of 18:36, 9 January 2008
Line 21:		Line 21:
	**c. How does this relate to Exercise 1? Does having the same PGF mean they are distributed the same?		**c. How does this relate to Exercise 1? Does having the same PGF mean they are distributed the same?

		+	* '''Exercise 4:''' Suppose that X has a pgf <math> P_{x}(t)=(1-p)+pt </math> and let <math> Y = aX + b.</math> What is <math> P_{y}(t) </math> ?

	==See also==		==See also==

@@ Line 23: / Line 23: @@
 ==See also==
-* [[SOCR_EduMaterials_FunctorActivities_MGF | Other SOCR Distribution Functor Activities]]
+* [[SOCR_EduMaterials_FunctorActivities_PGF | Other SOCR Distribution Functor Activities]]
 <hr>

← Older revision		Revision as of 06:30, 9 January 2008
Line 20:		Line 20:
	**b. Show that the PGF of the sum of <math>n</math> independent Geometric Random Variables with success probability <math> p </math> is the same as the MGF of the Negative-Binomial Distribution using the corollary above.		**b. Show that the PGF of the sum of <math>n</math> independent Geometric Random Variables with success probability <math> p </math> is the same as the MGF of the Negative-Binomial Distribution using the corollary above.
	**c. How does this relate to Exercise 1? Does having the same PGF mean they are distributed the same?		**c. How does this relate to Exercise 1? Does having the same PGF mean they are distributed the same?
		+
		+
		+	==See also==
		+	* [[SOCR_EduMaterials_FunctorActivities_MGF \| Other SOCR Distribution Functor Activities]]
		+
		+	<hr>
		+	* SOCR Home page: http://www.socr.ucla.edu
		+
		+	{{translate\|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_Binomial_PGF}}

@@ Line 1: / Line 1: @@
 == This is an activity to explore the Probability Generating Functions for the Bernoulli, Binomial, Geometric and Negative-Binomial Distributions.==
-* '''Description''':  You can access the applets for the above distributions at  http://www.socr.ucla.edu/htmls/SOCR_DistributionFunctors.html .
+* '''Description''':  You can access the applets for the above distributions at  http://socr.umich.edu/html/dist/ .
 * '''Exercise 1:''' Use SOCR to graph the PGF's and print the following distributions and answer the questions below.  Also, comment on the shape of each one of these distributions:

@@ Line 4: / Line 4: @@
 * '''Exercise 1:''' Use SOCR to graph the PGF's and print the following distributions and answer the questions below.  Also, comment on the shape of each one of these distributions:
-**a.<math> X \sim Bernoulli(0.5) </math>
+**a.<math> X \sim Bernoulli(0.1) </math>
-**b.<math> X \sim Binomial(1,0.5) </math>
+**b.<math> X \sim Binomial(10,0.9) </math>
-**c.<math> X \sim Geometric(0.5) </math>
+**c.<math> X \sim Geometric(0.3) </math>
-**d.<math> X \sim NegativeBinomial(1, 0.5) </math>
+**d.<math> X \sim NegativeBinomial(10, 0.7) </math>
 Below you can see a snapshot of the PGF of the distribution of <math> X \sim Bernoulli(0.8) </math>
@@ Line 14: / Line 14: @@
 Do you notice any similarities between the graphs of these PGF's between any of these distributions?
-* '''Exercise 2:''' Use SOCR to graph and print the PGF of the distribution of a geometric random variable with <math> p=0.2,  p=0.7 </math>.  What is the shape of this function?  What happens when <math> p </math> is large?  What happens when <math> p </math> is small?
+* '''Exercise 2:''' Use SOCR to graph and print the PGF of the distribution of a geometric random variable with <math> p=0.1,  p=0.8 </math>.  What is the shape of this function?  What happens when <math> p </math> is large?  What happens when <math> p </math> is small?
 * '''Exercise 3:''' You learned in class about the properties of PGF's If <math> X_1, ...X_n</math> are iid. and <math>Y = \sum_{i=1}^n X_i. </math> then <math>P_{y}(t) = {[P_{X_1}(t)]}^n</math>.

@@ Line 10: / Line 10: @@
 Below you can see a snapshot of the PGF of the distribution of <math> X \sim Bernoulli(0.8) </math>
-<center>[[Image:BernoulliPGF.jpg|600px]]</center>
+<center>[[Image:BernoulliPGF1.jpg|600px]]</center>
 Do you notice any similarities between the graphs of these PGF's between any of these distributions?
@@ Line 17: / Line 17: @@
 * '''Exercise 3:''' You learned in class about the properties of PGF's If <math> X_1, ...X_n</math> are iid. and <math>Y = \sum_{i=1}^n X_i. </math> then <math>P_{y}(t) = {[P_{X_1}(t)]}^n</math>.
-**a. Show that the PGF of the sum of <math>n</math> independent Bernoulli Trials with success probability <math> p </math> is the same as the MGF of the Binomial Distribution using the corollary above.
+**a. Show that the PGF of the sum of <math>n</math> independent Bernoulli Trials with success probability <math> p </math> is the same as the PGF of the Binomial Distribution using the corollary above.
 **b. Show that the PGF of the sum of <math>n</math> independent Geometric Random Variables with success probability <math> p </math> is the same as the MGF of the Negative-Binomial Distribution using the corollary above.
-**c.  How does this relate to Exercise 1?  Does having the same MGF mean they are distributed the same?
+**c.  How does this relate to Exercise 1?  Does having the same PGF mean they are distributed the same?