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://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?