SOCR EduMaterials Activities Discrete Distributions

From Socr

(Difference between revisions)

Revision as of 02:45, 22 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 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. 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, $P (X = x) = (1 - p) x - 1 p$ , $P (X > k) = (1 - p) k$ , 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.

Exericise 7: Binomial approximation to hypergeometric:

Let $X$ follow the hypergeometric probability distribution with Failed to parse (unknown function\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) . 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.

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

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@@ Line 4: / Line 4: @@
 * '''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.
+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.
-b. <math> X \sim b(10,0.1) <\math>,  find <math> P(1 \le X \le 3) <\math>.
+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>.
+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>.
+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>
+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?
+* '''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>
+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:
+* '''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>
+a. <math> P(X=5) </math>
-b. <math> P(X > 3) <\math>
+b. <math> P(X > 3) </math>
-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>
-e. <math> P(X \ge 8) <\math>
+e. <math> P(X \ge 8) </math>
-f. <math> P(4 \le X \le 9) <\math>
+f. <math> P(4 \le X \le 9) </math>
-g. <math> P(4 < X < 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 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.
+* '''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>
+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.
+* '''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.
+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 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 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=10 </math>, and number of ``hot" items 40 using the binomial probability distribution?  Explain.
@@ Line 43: / Line 43: @@
 * SOCR Home page: http://www.socr.ucla.edu
-{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ConfidenceIntervals}}</math>
+{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ConfidenceIntervals}}

SOCR EduMaterials Activities Discrete Distributions

From Socr

Revision as of 02:45, 22 October 2006

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

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