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. $X \sim Bernoulli(0.1)$
- 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 $X \sim Bernoulli(0.8)$

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 X₁,...X_n are iid. and then .
- 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 $P x (t) = (1 - p) + p t$ and let $Y = a X + b .$ What is $P y (t)$ ?

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