SOCR EduMaterials Activities Discrete Distributions
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. 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 \( N=1000, n=10 <\math> and number of ``hot" items 50. Graph and print this distribution. * '''Exercise 8:''' Refer to exerciise 7. Use SOCR to compute the exact probability'"`UNIQ-MathJax1-QINU`"'. Approximate <math> P(X=2) \) 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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