SOCR EduMaterials Activities Binomial Distributions

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Current revision as of 05:20, 15 October 2009

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. $X \sim b(10,0.5)$ , find $P (X = 3)$ , $E (X)$ , $s d (X)$ , 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 $X \sim b(20,0.3)$

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 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 $X \sim hypergeometric(N=100, n=15, r=30)$

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 $N=50, \ n=20$ , and number of "hot" items 40 using the binomial probability distribution? Graph and print the exact (hypergeometric) and the approximate (binomial) distributions and compare.

SOCR Home page: http://www.socr.ucla.edu

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@@ Line 44: / Line 44: @@
 * '''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 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: <math> P(X=2) </math>.  Approximate <math> P(X=2) </math> using the binomial distribution.  Is the approximation good?  Why?
+* '''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.
+* '''Exercise 9:''' Do you think you can approximate well the hypergeometric probability distribution with <math> N=50, \ n=20 </math>, and number of "hot" items 40 using the binomial probability distribution?  Graph and print the exact (hypergeometric) and the approximate (binomial) distributions and compare.

SOCR EduMaterials Activities Binomial Distributions

From Socr

Current revision as of 05:20, 15 October 2009

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

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