SOCR EduMaterials FunctorActivities Bernoulli Distributions

This is an activity to explore the Moment 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 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 \( X \sim Bernoulli(0.8) \)

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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 \( X_1, ...X_n\) 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 \(p\)