Difference between revisions of "SOCR CommunityPortal Events May2007"

From SOCR
Jump to: navigation, search
Line 10: Line 10:
  
  
 +
== This is an activity to explore the Binomial, Geometric, and Hypergeometric Probability Distributions.==
 +
 +
* '''Description''':  You can access the applets for the above distributions at  http://www.socr.ucla.edu/htmls/SOCR_Distributions.html .
 +
 +
* '''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:
 +
**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.
 +
**b. <math> X \sim b(10,0.1) </math>,  find <math> P(1 \le X \le 3) </math>. 
 +
**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>.
 +
**d. <math> X \sim b(30,0.1) </math>, find <math> P(X > 2) </math>.
 +
 +
Below you can see a snapshot of the distribution of <math> X \sim b(20,0.3) </math>
 +
 +
 +
<center>[[Image: SOCR_Activities_Binomial_Christou__binomial.jpg|600px]]</center>
 +
 +
 +
* '''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?
 +
 +
Below you can see a snapshot of the distribution of <math> X \sim geometric(0.4) </math>
 +
 +
 +
<center>[[Image: SOCR_Activities_Christou_geometric.jpg|600px]]</center>
 +
 +
<math>\sqrt(n)</math>
 +
 +
* '''Exercise 3:''' Select the geometric probability distribution with <math> p=0.2 </math>.  Use SOCR to compute the following:
 +
**a. <math> P(X=5) </math> 
 +
**b. <math> P(X > 3) </math>
 +
**c. <math> P(X \le 5) </math>
 +
**d. <math> P(X > 6) </math>
 +
**e. <math> P(X \ge 8) </math>
 +
**f. <math> P(4 \le X \le 9) </math>
 +
**g. <math> P(4 < X < 9) </math>
 +
 +
* '''Exercise 4:''' Verify that your answers in exercise 3 agree 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.
 +
 +
* '''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.
 +
 +
Below you can see a snapshot of the distribution of <math> X \sim hypergeometric(N=100, n=15, r=30) </math>
 +
 +
 +
<center>[[Image: SOCR_Activities_Christou_hypergeometric.jpg|600px]]</center>
 +
 +
 +
* '''Exercise 6:''' Refer to exercise 5.  Use SOCR to compute <math> P(X=5) </math> and write down the formula that gives this answer.
 +
 +
* '''Exercise 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.
 +
 +
* '''Exercise 8:''' Refer to exercise 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?
 +
 +
* '''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.
 +
 +
 +
<hr>
 +
* SOCR Home page: http://www.socr.ucla.edu
 +
 +
{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_CommunityPortal_Events_May2007}}
  
  

Revision as of 15:19, 18 May 2007

SOCR Events-Specific Community Pages SOCR/USCOTS May 2007 Pages

Please try to keep these pages as clean and hierarchically organized as possible. Refer to the SOCR Editing Guide before you begin contributing to these resources.




This is an activity to explore the Binomial, Geometric, and Hypergeometric Probability Distributions.

  • 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:
    • a. \( X \sim b(10,0.5) \), find \( P(X=3) \), \( E(X) \), \( sd(X) \), and verify them with the formulas discussed in class.
    • b. \( X \sim b(10,0.1) \), find \( P(1 \le X \le 3) \).
    • c. \( X \sim b(10,0.9) \), find \( P(5 < X < 8), \ P(X < 8), \ P(X \le 7), \ P(X \ge 9) \).
    • d. \( X \sim b(30,0.1) \), find \( P(X > 2) \).

Below you can see a snapshot of the distribution of \( X \sim b(20,0.3) \)


SOCR Activities Binomial Christou binomial.jpg


  • Exercise 2: Use SOCR to graph and print the distribution of a geometric random variable with \( p=0.2, p=0.7 \). What is the shape of these distributions? What happens when \( p \) is large? What happens when \( p \) is small?

Below you can see a snapshot of the distribution of \( X \sim geometric(0.4) \)


SOCR Activities Christou geometric.jpg

\(\sqrt(n)\)

  • Exercise 3: Select the geometric probability distribution with \( p=0.2 \). Use SOCR to compute the following:
    • a. \( P(X=5) \)
    • b. \( P(X > 3) \)
    • c. \( P(X \le 5) \)
    • d. \( P(X > 6) \)
    • e. \( P(X \ge 8) \)
    • f. \( P(4 \le X \le 9) \)
    • g. \( P(4 < X < 9) \)
  • Exercise 4: Verify that your answers in exercise 3 agree with the formulas discussed in class, for example, \( P(X=x)=(1-p)^{x-1}p \), \( P(X > k)=(1-p)^k \), etc. Write all your answers in detail using those formulas.
  • Exercise 5: Let \( X \) follow the hypergeometric probability distribution with \( N=52 \), \( n=10 \), and number of "hot" items 13. Use SOCR to graph and print this distribution.

Below you can see a snapshot of the distribution of \( X \sim hypergeometric(N=100, n=15, r=30) \)


SOCR Activities Christou hypergeometric.jpg


  • Exercise 6: Refer to exercise 5. Use SOCR to compute \( P(X=5) \) and write down the formula that gives this answer.
  • Exercise 7: Binomial approximation to hypergeometric: Let \( X \) follow the hypergeometric probability distribution with \( N=1000, \ n=10 \) and number of "hot" items 50. Graph and print this distribution.
  • Exercise 8: Refer to exercise 7. Use SOCR to compute the exact probability\[ P(X=2) \]. Approximate \( P(X=2) \) using the binomial distribution. Is the approximation good? Why?
  • Exercise 9: Do you think you can approximate well the hypergeometric probability distribution with \( N=50, \ n=10 \), and number of "hot" items 40 using the binomial probability distribution? Explain.


Translate this page:

(default)
Uk flag.gif

Deutsch
De flag.gif

Español
Es flag.gif

Français
Fr flag.gif

Italiano
It flag.gif

Português
Pt flag.gif

日本語
Jp flag.gif

България
Bg flag.gif

الامارات العربية المتحدة
Ae flag.gif

Suomi
Fi flag.gif

इस भाषा में
In flag.gif

Norge
No flag.png

한국어
Kr flag.gif

中文
Cn flag.gif

繁体中文
Cn flag.gif

Русский
Ru flag.gif

Nederlands
Nl flag.gif

Ελληνικά
Gr flag.gif

Hrvatska
Hr flag.gif

Česká republika
Cz flag.gif

Danmark
Dk flag.gif

Polska
Pl flag.png

România
Ro flag.png

Sverige
Se flag.gif






Translate this page:

(default)
Uk flag.gif

Deutsch
De flag.gif

Español
Es flag.gif

Français
Fr flag.gif

Italiano
It flag.gif

Português
Pt flag.gif

日本語
Jp flag.gif

България
Bg flag.gif

الامارات العربية المتحدة
Ae flag.gif

Suomi
Fi flag.gif

इस भाषा में
In flag.gif

Norge
No flag.png

한국어
Kr flag.gif

中文
Cn flag.gif

繁体中文
Cn flag.gif

Русский
Ru flag.gif

Nederlands
Nl flag.gif

Ελληνικά
Gr flag.gif

Hrvatska
Hr flag.gif

Česká republika
Cz flag.gif

Danmark
Dk flag.gif

Polska
Pl flag.png

România
Ro flag.png

Sverige
Se flag.gif

Retrieved from "https://wiki.socr.umich.edu/index.php?title=SOCR_CommunityPortal_Events_May2007&oldid=3586"