SOCR EduMaterials Activities Discrete Distributions

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

  • Description: You can access the applets for the above distributions at [1]
  • 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. 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>. 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>

  • 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>

  • 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> b. <math> P(X > 3) <\math> c. <math> P(X \le 5) <\math> d. <math> P(X > 6) <\math> e. <math> P(X \ge 8) <\math> f. <math> P(4 \le X \le 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 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>

  • 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 8: Refer to exerciise 7. Use SOCR to compute the exact probability\[ 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? 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