SOCR - User contributions [en]

SOCR EduMaterials Activities Binomial PGF

2008-01-09T18:36:11Z

Rgidwani:

== 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(0.1) </math>
**b.<math> X \sim Binomial(10,0.9) </math>
**c.<math> X \sim Geometric(0.3) </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>
<center>[[Image:BernoulliPGF1.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=0.1, 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>.
**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 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==
* [[SOCR_EduMaterials_FunctorActivities_PGF | Other SOCR Distribution Functor Activities]]

<hr>
* SOCR Home page: http://www.socr.ucla.edu

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SOCR EduMaterials Activities Binomial PGF

2008-01-09T07:03:50Z

Rgidwani:

== 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(0.1) </math>
**b.<math> X \sim Binomial(10,0.9) </math>
**c.<math> X \sim Geometric(0.3) </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>
<center>[[Image:BernoulliPGF1.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=0.1, 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>.
**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 PGF mean they are distributed the same?

==See also==
* [[SOCR_EduMaterials_FunctorActivities_PGF | Other SOCR Distribution Functor Activities]]

<hr>
* SOCR Home page: http://www.socr.ucla.edu

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File:BernoulliPGF1.jpg

2008-01-09T03:40:30Z

Rgidwani:

SOCR EduMaterials Activities Binomial PGF

2008-01-09T03:39:21Z

Rgidwani:

== 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:BernoulliPGF1.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 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 PGF mean they are distributed the same?

File:BernoulliPGF.jpg

2008-01-09T03:37:52Z

Rgidwani:

SOCR EduMaterials Activities Binomial PGF

2008-01-09T03:37:25Z

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...

== 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?

SOCR EduMaterials FunctorActivities PGF

2008-01-09T03:33:08Z

Rgidwani:

== [[SOCR_EduMaterials_FunctorActivities | SOCR Functor Activities]] - Probability Generating Function (PGF) Activities ==

* [[Help_pages_for_SOCR_Functors]]
* [[About_pages_for_SOCR_Functors]]
<hr>

* [[SOCR_EduMaterials_Activities_Binomial_PGF | SOCR Bernoulli, Binomial, Geometric and Negative-Binomial PGF Activity]]

<hr>
* SOCR Home page: http://www.socr.ucla.edu

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SOCR EduMaterials FunctorActivities MGF Moments

2008-01-09T03:31:18Z

Rgidwani:

== This is an activity to explore useful properties of MGF's.==

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

* '''Exercise 1:''' As you have learned in class, there are quite a few interesting properties that Moment Generating Functions hold. For example you learned that <math> E(X^n)=M_{x}^{(n)}(0)={d^n M_x(t)\over{dt^n}}\mid_{t=0} </math> If the MGF is defined in the neighborhood of 0. So to get the Expected Value for a particular distribution, you would take the first derivative of the MGF and set t=0. Use SOCR to graph and print the following distributions and answer the questions below. ''You must do these exercises using MGF's, you can find the slope using the mouse pointer.''
**a. Find the Expected Value of <math> X \sim Binomial(10,.5) </math>
**b. Find the Expected Value of <math> X \sim Normal(0,1) </math>
**c. Find the Expected Value of <math> X \sim ChiSquare(13) </math>

* '''Exercise 2:''' Can you use MGF's to find the Expected Value for the Continuous Uniform Distribution? Why or why not?

* '''Exercise 3:''' In Exercise 1, we calculated the <math>1^{st}</math> Moment. If we take the second derivative of the MGF with respect to t, where <math> t=0 </math>. We get <math> E(X^2) </math>. We can use this to find the Variance of a particular Distribution. Repeat Parts (a,b,c) for Exercise 1, but this time calculate the variance.

* '''Exercise 4:''' What do we get when we take the <math>3^{rd}</math> and <math<4^{th}</math> derivatives of a MGF and set <math> t=0 </math>?

SOCR EduMaterials FunctorActivities MGF Moments

2008-01-09T03:28:40Z

Rgidwani:

== This is an activity to explore useful properties of MGF's.==

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

* '''Exercise 1:''' As you have learned in class, there are quite a few interesting properties that Moment Generating Functions hold. For example you learned that <math> E(X^n)=M_{x}^{(n)}(0)={d^n M_x(t)\over{dt^n}}\mid_{t=0} </math> If the MGF is defined in the neighborhood of 0. So to get the Expected Value for a particular distribution, you would take the first derivative of the MGF and set t=0. Use SOCR to graph and print the following distributions and answer the questions below. ''You must do these exercises using MGF's, you can find the slope using the mouse pointer.''
**a. Find the Expected Value of <math> X \sim Binomial(10,.5) </math>
**b. Find the Expected Value of <math> X \sim Normal(0,1) </math>
**c. Find the Expected Value of <math> X \sim ChiSquare(13) </math>

* '''Exercise 2:''' Can you use MGF's to find the Expected Value for the Continuous Uniform Distribution? Why or why not?

* '''Exercise 3:''' In Exercise 1, we calculated the 1^{st} Moment. If we take the second derivative of the MGF with respect to t, where <math> t=0 </math>. We get <math> E(X^2) </math>. We can use this to find the Variance of a particular Distribution. Repeat Parts (a,b,c) for Exercise 1, but this time calculate the variance.

* '''Exercise 4:''' What do we get when we take the 3^{rd} and 4^{th} derivatives of a MGF and set <math> t=0 </math>?

SOCR EduMaterials FunctorActivities MGF Moments

2008-01-09T03:11:48Z

Rgidwani: New page: == This is an activity to explore useful properties of MGF's.== * '''Description''': You can access the applets for the above distributions at http://www.socr.ucla.edu/htmls/SOCR_Distri...

SOCR EduMaterials FunctorActivities MGF

2008-01-09T03:06:50Z

Rgidwani:

== [[SOCR_EduMaterials_FunctorActivities | SOCR Functor Activities]] - Moment Generating Function (MGF) Activities ==

* [[SOCR_EduMaterials_FunctorActivities_Bernoulli_Distributions | SOCR Bernoulli, Binomial, Geometric and Negative-Binomial MGF Activity]]
* [[SOCR_EduMaterials_FunctorActivities_MGF_Moments | SOCR Using MGFs to get Moments Activity]]

<hr>
* SOCR Home page: http://www.socr.ucla.edu

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SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T03:02:35Z

Rgidwani:

== This is an activity to explore the Moment 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 MGF'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 MGF of the distribution of <math> X \sim Bernoulli(0.8) </math>
<center>[[Image:BernoulliMGF.jpg|600px]]</center>

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

* '''Exercise 2:''' Use SOCR to graph and print the MGF 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 MGF's If <math> X_1, ...X_n</math> are iid. and <math>Y = \sum_{i=1}^n X_i. </math> then <math>M_{y}(t) = {[M_{X_1}(t)]}^n</math>.
**a. Show that the MGF 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 MGF 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?

* '''Exercise 4:''' Graph the PDF and the MGF for the appropriate Distribution where <math> M_x(t)={({3 \over 4}e^t+{1\over 4})}^{15} </math>. Be sure to include the correct parameters for this distribution, for example if <math> X \sim Geometric(p) </math> be sure to include the numeric value for <math>p</math>

SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T03:02:05Z

Rgidwani:

SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T02:54:06Z

Rgidwani:

== This is an activity to explore the Moment 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 MGF'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 MGF of the distribution of <math> X \sim Bernoulli(0.8) </math>
<center>[[Image:BernoulliMGF.jpg|600px]]</center>

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

* '''Exercise 2:''' Use SOCR to graph and print the MGF 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 MGF's If <math> X_1, ...X_n</math> are iid. and <math>Y = \sum_{i=1}^n X_i. </math> then <math>M_{y}(t) = {[M_{X_1}(t)]}^n</math>.
**a. Show that the MGF of the sum of n 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 MGF of the sum of n 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?

SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T02:51:38Z

Rgidwani:

== This is an activity to explore the Moment 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 MGF'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 MGF of the distribution of <math> X \sim Bernoulli(0.8) </math>
<center>[[Image:BernoulliMGF.jpg|600px]]</center>

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

* '''Exercise 2:''' Use SOCR to graph and print the MGF 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 MGF's If <math> X_1, ...X_n</math> are iid. and <math>Y = \sum_{i=1}^n X_i. </math> then <math>M_{y}(t) = {[M_{X_1}(t)]}^n</math>.
**a. Show that the MGF of the sum of n independent Bernoulli Trials with success probability <math> p </math> is the same as the MGF of the Binomial Distribution using the corollar above.
**b. Show that the MGF of the sum of n independent Geometric

SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T02:48:13Z

Rgidwani:

SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T02:44:29Z

Rgidwani:

SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T02:29:57Z

Rgidwani:

File:BernoulliMGF.jpg

2008-01-09T02:28:39Z

Rgidwani:

SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T02:28:11Z

Rgidwani:

SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T02:25:22Z

Rgidwani:

SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T02:20:45Z

Rgidwani:

SOCR EduMaterials FunctorActivities Bernoulli Distributions

2008-01-09T02:14:38Z

Rgidwani: New page: == This is an activity to explore the Moment Generating Functions for the Bernoulli, Binomial, Geometric and Negative-Binomial Distributions.== * '''Description''': You can access the ap...

SOCR EduMaterials FunctorActivities MGF

2008-01-08T20:34:16Z

Rgidwani:

== [[SOCR_EduMaterials_FunctorActivities | SOCR Functor Activities]] - Moment Generating Function (MGF) Activities ==

* [[SOCR_EduMaterials_FunctorActivities_Bernoulli_Distributions | SOCR Bernoulli Binomial MGF Activity]]

* [[About_pages_for_SOCR_Functors_MGF]]
* [[Help_pages_for_SOCR_Functors_MGF]]
<hr>

* TBD

<hr>
* SOCR Home page: http://www.socr.ucla.edu

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UQuadraticDistribuionAbout

2007-11-08T20:09:24Z

Rgidwani: Updated MGF and CF

== [[About_pages_for_SOCR_Distributions]] - U-Quadratic Distribution ==

===Description===
[[Image:SOCR_Distributions_UQuadraticAbout_Dinov_Fig1.jpg|400px|thumbnail|right]]
The ''U quadratic distribution'' is defined by the following density function
<center><math> f(x)=\alpha \left ( x - \beta \right )^2, \forall x \in [a , b], a < b</math>, </center>
where the relation between the two pairs of parameters (domain-support, a and b) and (range/offset <math>\alpha</math> and <math>\beta</math>) are given by the following two equations
<center>(gravitational balance center) <math>\beta = {b+a \over 2}</math>, and
</center>
<center>(vertical scale) <math>\alpha = {12 \over \left ( b-a \right )^3}</math>.
</center>
More information about U-quadratic, and other continuous distribution functions, is available at [http://en.wikipedia.org/wiki/UQuadratic_distribution Wikipedia].

===Properties===
* Support Parameters: <math>a < b \in (-\infty,\infty)</math>
* Scale/Offset Parameters: <math>\alpha \in (0,\infty)</math> and <math>\beta \in (-\infty,\infty)</math>
* PDF: <math>f(x)=\alpha \left ( x - \beta \right )^2, \forall x \in [a , b]</math>
* CDF <math>F(x)={\alpha \over 3} \left ( (x - \beta)^3 + (\beta - a)^3 \right ), \forall x \in [a , b]</math>
* Mean: <math>{a+b \over 2}</math>
* Median: <math>{a+b \over 2}</math>
* Modes: <math>a </math> and <math> b </math>
* Variance: <math> {3 \over 20} (b-a)^2 </math>
* Skewness: 0 (distribution is symmetric around the mean)
* Kurtosis: <math> {3 \over 112} (b-a)^4 </math>
* Moment Generating Function: <math>M_x(t)= {-3\left(e^{at}(4+(a^2+2a(-2+b)+b^2)t)- e^{bt} (4 + (-4b + (a+b)^2)t)\right) \over (a-b)^3 t^2 }</math>
* Characteristic Function: <math>{3i\left(e^{iat}(-4i+(a^2+2a(-2+b)+b^2)t)+ e^{ibt} (4i - (-4b + (a+b)^2)t)\right) \over (a-b)^3 t^2 }</math>

===Interactive U Quadratic Distribution===
You can see the interactive ''U Quadratic'' distribution by going to [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions] and selecting from the drop down list of distributions ''U Quadratic''. Then follow the '''Help''' instructions to dynamically set parameters, compute critical and probability values using the mouse and keyboard.

<center>[[Image:SOCR_Distributions_UQuadraticAbout_Dinov_Fig2.jpg|500px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

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