# SOCR CommunityPortal Events May2007

(Difference between revisions)
 Revision as of 15:42, 18 April 2007 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 20:13, 18 May 2007 (view source)IvoDinov (Talk | contribs) Newer edit → 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. $X \sim b(10,0.5)$, 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 $X \sim b(20,0.3)$ + + +
[[Image: SOCR_Activities_Binomial_Christou__binomial.jpg|600px]]
+ + + * '''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 $X \sim geometric(0.4)$ + + +
[[Image: SOCR_Activities_Christou_geometric.jpg|600px]]
+ + $\sqrt(n)$ + + * '''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 $X \sim hypergeometric(N=100, n=15, r=30)$ + + +
[[Image: SOCR_Activities_Christou_hypergeometric.jpg|600px]]
+ + + * '''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. + + +

## 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 $X \sim b(20,0.3)$

• 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 $X \sim geometric(0.4)$

$\sqrt(n)$

• 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 − 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.
• 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.