SOCR EduMaterials FunctorActivities Bernoulli Distributions

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Revision as of 02:45, 9 January 2008

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 $X \sim Bernoulli(0.8)$

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,...X_n are iid. and then .
- 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 corollar above.
- b. Show that the MGF of the sum of n independent Geometric

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 * '''Exercise 3:''' You learned in class about the properties of MGF's If <math> X_1, ...X_n</math> are iid. and <math>Y = \sum_{i=1}^n X_i. </math> then <math>M_{y}(t) = {[M_{X_1}(t)]}^n</math>.
+**a. Show that the MGF of the sum of n independent Bernoulli Trials with success probability <math> p </math> is the same as the MGF of the Binomial Distribution using the corollar above.
+**b. Show that the MGF of the sum of n independent Geometric