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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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? | ||
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* '''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? |
Revision as of 22:39, 8 January 2008
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.\( X \sim Bernoulli(0.5) \)
- b.\( X \sim Binomial(1,0.5) \)
- c.\( X \sim Geometric(0.5) \)
- d.\( X \sim NegativeBinomial(1, 0.5) \)
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 \( p=0.2, p=0.7 \). 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?
