SOCR EduMaterials Activities Binomial PGF
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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