# SOCR EduMaterials Activities Binomial PGF

## This is an activity to explore the Probability Generating Functions for the Bernoulli, Binomial, Geometric and Negative-Binomial 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.$X \sim Bernoulli(0.1)$
• 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,...Xn 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 Px(t) = (1 − p) + pt and let Y = aX + b. What is Py(t) ?