SOCR EduMaterials FunctorActivities Bernoulli Distributions

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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

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