SOCR EduMaterials Activities Discrete Distributions

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(→This is an activity to explore the Binomial, Geometric, and Hypergeometric Probability Distributions.)
(→This is an activity to explore the Binomial, Geometric, and Hypergeometric Probability Distributions.)
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== This is an activity to explore the Binomial, Geometric, and Hypergeometric Probability Distributions.==
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== This is an activity to explore the Negaive Binomial Probability Distribution.==
* '''Description''':  You can access the applets for the above distributions at  http://www.socr.ucla.edu/htmls/SOCR_Distributions.html .  
* '''Description''':  You can access the applets for the above distributions at  http://www.socr.ucla.edu/htmls/SOCR_Distributions.html .  
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* '''Exercise 1:''' Use SOCR to graph and print the following distributions and answer the questions below.  Also, comment on the shape of each one of these distributions:  
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* '''Exercise 1:''' Use SOCR to graph and print the following distributions and comment on the shape of each one of them:
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**a. <math> X \sim b(10,0.5) </math>, find <math> P(X=3) </math>, <math> E(X) </math>, <math> sd(X) </math>, and verify them with the formulas discussed in class.
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** a. <math> X <\math> follows negative binomial with <math> r=5, \ p=0.2 <\math>. 
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**b. <math> X \sim b(10,0.1) </math>,  find <math> P(1 \le X \le 3) </math>.
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** b. <math> X <\math> follows negative binomial with <math> r=5, \ p=0.9 <\math>.  
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**c. <math> X \sim b(10,0.9) </math>, find <math> P(5 < X < 8), \ P(X < 8), \ P(X \le 7), \ P(X \ge 9) </math>.
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** c. <math> X <\math> follows negative binomial with <math> r=20, \ p=0.3 <\math>.  
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**d. <math> X \sim b(30,0.1) </math>, find <math> P(X > 2) </math>.
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** d. <math> X <\math> follows negative binomial with <math> r=20, \ p=0.9 <\math>.
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* '''Exercise 2:''' Let <math> X <\math>  follows the negative binomial distribution with <math> r=5, \ p=0.2 <\math>.  Explain in words what <math> P(X > 10) <\math> means and use SOCR to compute this probability.
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Below you can see a snapshot of the distribution of <math> X \sim b(20,0.3) </math>
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Below you can see the distribution of <math> X \sim b(20,0.3) <\math>
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<center>[[Image: SOCR_Activities_Binomial_Christou__binomial.jpg|600px]]</center>
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<center>[[Image: SOCR_Activities_Christou_negbinomial.jpg|600px]]</center>
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* '''Exercise 2:''' Use SOCR to graph and print the distribution of a geometric random variable with <math> p=0.2,  p=0.7 </math>.  What is the shape of these distributions?  What happens when <math> p </math> is large?  What happens when <math> p </math> is small?
 
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Below you can see a snapshot of the distribution of <math> X \sim geometric(0.4) </math>
 
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<center>[[Image: SOCR_Activities_Christou_geometric.jpg|600px]]</center>
 
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* '''Exercise 3:''' Select the geometric probability distribution with <math> p=0.2 </math>.  Use SOCR to compute the following:
 
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**a. <math> P(X=5) </math> 
 
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**b. <math> P(X > 3) </math>
 
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**c. <math> P(X \le 5) </math>
 
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**d. <math> P(X > 6) </math>
 
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**e. <math> P(X \ge 8) </math>
 
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**f. <math> P(4 \le X \le 9) </math>
 
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**g. <math> P(4 < X < 9) </math>
 
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* '''Exercise 4:''' Verify that your answers in exercise 3agree with the formulas discussed in class, for example, <math> P(X=x)=(1-p)^{x-1}p </math>, <math> P(X > k)=(1-p)^k </math>, etc.  Write all your answers in detail using those formulas.
 
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* '''Exercise 5:''' Let <math> X </math> follow the hypergeometric probability distribution with <math> N=52 </math>, <math> n=10 </math>, and number of "hot" items 13.  Use SOCR to graph and print this distribution.
 
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Below you can see a snapshot of the distribution of <math> X \sim hypergeometric(N=100, n=15, r=30) </math>
 
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<center>[[Image: SOCR_Activities_Christou_hypergeometric.jpg|600px]]</center>
 
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* '''Exercise 6:''' Refer to exercise 5.  Use SOCR to compute <math> P(X=5) </math> and write down the formula that gives this answer.
 
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* '''Exericise 7:''' Binomial approximation to hypergeometric:  Let <math> X </math> follow the hypergeometric probability distribution with <math> N=1000, \ n=10 </math> and number of "hot" items 50.  Graph and print this distribution.
 
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* '''Exercise 8:''' Refer to exerciise 7.  Use SOCR to compute the exact probability: <math> P(X=2) </math>.  Approximate <math> P(X=2) </math> using the binomial distribution.  Is the approximation good?  Why?
 
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* '''Exercise 9:''' Do you think you can approximate well the hypergeometric probability distribution with <math> N=50, \ n=10 </math>, and number of "hot" items 40 using the binomial probability distribution?  Explain.
 
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</math>

Revision as of 22:48, 27 October 2006

This is an activity to explore the Negaive Binomial Probability Distribution.

  • Exercise 1: Use SOCR to graph and print the following distributions and comment on the shape of each one of them:
    • a. Failed to parse (unknown function\math): X <\math> follows negative binomial with <math> r=5, \ p=0.2 <\math>. ** b. <math> X <\math> follows negative binomial with <math> r=5, \ p=0.9 <\math>. ** c. <math> X <\math> follows negative binomial with <math> r=20, \ p=0.3 <\math>. ** d. <math> X <\math> follows negative binomial with <math> r=20, \ p=0.9 <\math>. * '''Exercise 2:''' Let <math> X <\math> follows the negative binomial distribution with <math> r=5, \ p=0.2 <\math>. Explain in words what <math> P(X > 10) <\math> means and use SOCR to compute this probability. Below you can see the distribution of <math> X \sim b(20,0.3) <\math> <center>[[Image: SOCR_Activities_Christou_negbinomial.jpg|600px]]</center> <hr> * SOCR Home page: http://www.socr.ucla.edu {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ConfidenceIntervals}}
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