SOCR EduMaterials Activities Discrete Distributions

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* '''Exercise 3:''' Select the geometric probability distribution with <math> p=0.2 </math>.  Use SOCR to compute the following:
* '''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>   
+
*a. <math> P(X=5) </math>   
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b. <math> P(X > 3) </math>  
+
*b. <math> P(X > 3) </math>  
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c. <math> P(X \le 5) </math>
+
*c. <math> P(X \le 5) </math>
d. <math> P(X > 6) </math>
d. <math> P(X > 6) </math>

Revision as of 02:50, 22 October 2006

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. b.  X \sim b(10,0.1) , find  P(1 \le X \le 3) . c.  \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)

  • 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)

  • 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 3agree with the formulas we discussed in class, for example, P(X = x) = (1 − p)x − 1p, 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)

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





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