Difference between revisions of "SOCR EduMaterials Activities Binomial PGF"
(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...) |
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== This is an activity to explore the Probability Generating Functions for the Bernoulli, Binomial, Geometric and Negative-Binomial Distributions.== | == 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:// | + | * '''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: | * '''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. | + | **a.<math> X \sim Bernoulli(0.1) </math> |
| − | **b.<math> X \sim Binomial( | + | **b.<math> X \sim Binomial(10,0.9) </math> |
| − | **c.<math> X \sim Geometric(0. | + | **c.<math> X \sim Geometric(0.3) </math> |
| − | **d.<math> X \sim NegativeBinomial( | + | **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> | Below you can see a snapshot of the PGF of the distribution of <math> X \sim Bernoulli(0.8) </math> | ||
| − | <center>[[Image: | + | <center>[[Image:BernoulliPGF1.jpg|600px]]</center> |
Do you notice any similarities between the graphs of these PGF's between any of these distributions? | 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. | + | * '''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>. | * '''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 | + | **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. | **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 | + | **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.umich.edu | ||
| + | |||
| + | {{translate|pageName=http://wiki.socr.umich.edu/index.php?title=SOCR_EduMaterials_Activities_Binomial_PGF}} | ||
Latest revision as of 09:55, 5 March 2014
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://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:
- a.\( X \sim Bernoulli(0.1) \)
- b.\( X \sim Binomial(10,0.9) \)
- c.\( X \sim Geometric(0.3) \)
- d.\( X \sim NegativeBinomial(10, 0.7) \)
Below you can see a snapshot of the PGF of the distribution of \( X \sim Bernoulli(0.8) \)
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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 \( p=0.1, p=0.8 \). What is the shape of this function? What happens when \( p \) is large? What happens when \( p \) is small?
- Exercise 3: You learned in class about the properties of PGF's If \( X_1, ...X_n\) are iid. and \(Y = \sum_{i=1}^n X_i. \) then \(P_{y}(t) = {[P_{X_1}(t)]}^n\).
- a. Show that the PGF of the sum of \(n\) independent Bernoulli Trials with success probability \( p \) is the same as the PGF of the Binomial Distribution using the corollary above.
- b. Show that the PGF of the sum of \(n\) independent Geometric Random Variables with success probability \( p \) 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 \( P_{x}(t)=(1-p)+pt \) and let \( Y = aX + b.\) What is \( P_{y}(t) \) ?
See also
- SOCR Home page: http://www.socr.umich.edu
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