# SOCR EduMaterials FunctorActivities Bernoulli Distributions

## This is an activity to explore the Moment Generating Functions for the Bernoulli, Binomial, Geometric and Negative-Binomial Distributions.

• Exercise 1: Use SOCR to graph the MGF'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 MGF of the distribution of $X \sim Bernoulli(0.8)$

Do you notice any similarities between the graphs of these MGF's between any of these distributions?

• Exercise 2: Use SOCR to graph and print the MGF 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 MGF's If X1,...Xn are iid. and $Y = \sum_{i=1}^n X_i.$ then $M_{y}(t) = {[M_{X_1}(t)]}^n$.
• a. Show that the MGF of the sum of n independent Bernoulli Trials with success probability p is the same as the MGF of the Binomial Distribution using the corollary above.
• b. Show that the MGF 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 MGF mean they are distributed the same?
• Exercise 4: Graph the PDF and the MGF for the appropriate Distribution where $M_x(t)={({3 \over 4}e^t+{1\over 4})}^{15}$. Be sure to include the correct parameters for this distribution, for example if $X \sim Geometric(p)$ be sure to include the numeric value for p