Difference between revisions of "SOCR EduMaterials Activities Discrete Distributions"

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== This is an activity to explore the Binomial, Geometric, and Hypergeometric Probability Distributions.==
  
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\item[a.]  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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* '''Description''':  You can access the applets for the above distributions at [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html]  
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$X \sim b(10,0.5)$, find $P(X=3), E(X), sd(X)$, and verify them with the formulas.  \\
 
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$X \sim b(10,0.1)$,  find $P(1 \le X \le 3)$.  \\
 
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$X \sim b(10,0.9)$, find $P(5 < X < 8), P(X < 8), P(X \le 7), P(X \ge 9)$.  \\
 
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$X \sim b(30,0.1)$, find $P(X > 2)$.
 
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%Below you can see the distribution of <math> X \sim b(20,0.3) <\math>
 
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\item[b.]  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?
 
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%Below you can see the distribution of <math> X \sim geometric(0.4) <\math>
 
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\item[c.]  Select the geometric probability distribution with $p=0.2$.  Use $SOCR$ to compute the following:
 
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\begin{itemize}
 
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\item[1.] $P(X=5)$
 
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\item[2.] $P(X > 3)$
 
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\item[3.] $P(X \le 5)$
 
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\item[4.]  $P(X > 6)$
 
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\item[5.]  $P(X \ge 8)$
 
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\item[6.]  $P(4 \le X \le 9)$
 
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\item[7.] $P(4 < X < 9)$
 
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\end{itemize}
 
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\item[d.] Verify that your answers in part (c) agree with the formulas we 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.
 
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\item[e.] 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.
 
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%Below you can see the distribution of <math> X \sim hypergeometric(N=100, n=15, r=30) <\math>
 
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\item[f.] Refer to part (e).  Use $SOCR$ to compute $P(X=5)$ and write down the formula that gives this answer.
 
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\item[g.] Binomial approximation to hypergeometric: \\
 
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Let $X$ follow the hypergeometric probability distribution with $N=1000, n=10$ and number of ``hot" items 50.  Graph and print this distribution.
 
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\item[h.] Refer to part (g). Use $SOCR$ to compute the exact probability: $P(X=2)$. Approximate $P(X=2)$ using binomial distribution. Is the approximation good?  Why?
 
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\item[i.] 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.
 
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\end{itemize}
 
  
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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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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.
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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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c. <math> \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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d. <math> X \sim b(30,0.1) <\math>, find <math> P(X > 2) <\math>.
  
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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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* '''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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* '''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 we 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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* '''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:
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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.
  
  

Revision as of 21:33, 21 October 2006

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

  • Description: You can access the applets for the above distributions at [1]
  • 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 <math> P(X=3) <\math>, <math> E(X) <\math>, <math> sd(X) <\math>, and verify them with the formulas. b. <math> X \sim b(10,0.1) <\math>, find <math> P(1 \le X \le 3) <\math>. c. <math> \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>

  • 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>

  • 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 3agree with the formulas we 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>

  • Exercise 6: Refer to exercise 5. Use SOCR to compute <math> P(X=5) <\math> and write down the formula that gives this answer.
  • 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.

  • Exercise 8: Refer to exerciise 7. Use SOCR to compute the exact probability\[ 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? 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