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Discrete Random Variables: Lab I

Module by: Barbara Illowsky, Ph.D., Susan Dean. E-mail the authors

Summary: This module allows students to explore concepts related to discrete random variables through the use of a simple playing card experiment. Students will compare empirical data to a theoretical distribution to determine if the experiment fist a discrete distribution. This lab involves the concept of long-term probabilities. Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.

Note: You are viewing an old version of this document. The latest version is available here.

Class Time:

Names:

Student Learning Outcomes:

  • The student will compare empirical data and a theoretical distribution to determine if everyday experiment fits a discrete distribution.
  • The student will demonstrate an understanding of long-term probabilities.

Supplies:

  • One full deck of playing cards

Do the Experiment:

    • The experiment is to pick one card from a deck of shuffled cards.
    • The theoretical probability of picking a diamond from a deck is: _______
    • Shuffle a deck of cards. Pick one card from it. Record whether it was a diamond or not a diamond. Put the card back and reshuffle. Do this a total of 10 times. Record the number of diamonds picked: _______
    • Let XX size 12{X} {} = number of diamonds. Theoretically, X~X~ size 12{X "~" } {} ­­­_____________
  1. Record the number of diamonds picked for your class in the chart below. Then calculate the relative frequency.
    Table 1
    X X size 12{X} {} frequency   relative frequency
    0      
           
    1      
           
    2      
           
    3      
           
    4      
           
    5      
           
    6      
           
    7      
           
    8      
           
    9      
           
    10      
    • x¯=x¯= size 12{ {overline {x}} ={}} {} ­­
    • s=s= size 12{s={}} {}
  2. Construct a histogram of the empirical data.
    Figure 1
    Figure 1 (graph1.PNG)
  3. Build the theoretical PDF chart for X based on the distribution in Part I.
    Table 2
    X X size 12{X} {} P X = x P X = x size 12{P left (X=x right )} {}
    0  
       
    1  
       
    2  
       
    3  
       
    4  
       
    5  
       
    6  
       
    7  
       
    8  
       
    9  
       
    10  
    • μ=μ= size 12{μ={}} {}
    • σ=σ= size 12{σ={}} {}
  4. Construct a histogram of the theoretical distribution.
    Figure 2
    Figure 2 (graph2.PNG)
  5. Calculate the following, rounding to 4 decimal places:

    Use Part III here:

    • P X = 3 = P X = 3 = size 12{P left (X=3 right )={}} {}
    • P 1 < X < 4 = P 1 < X < 4 = size 12{P left (1<X<4 right )={}} {}
    • P X 8 = P X 8 = size 12{P left (X >= 8 right )={}} {}

    Note:

    RFRF size 12{ ital "RF"} {}= relative frequency

    Use Part II here:

    • RF X = 3 = RF X = 3 = size 12{ ital "RF" left (X=3 right )={}} {}
    • RF 1 < X < 4 = RF 1 < X < 4 = size 12{ ital "RF" left (1<X<4 right )={}} {}
    • RF X 8 = RF X 8 = size 12{ ital "RF" left (X >= 8 right )={}} {}
  6. Knowing that data vary, describe three similarities between the graphs and distributions of the theoretical and empirical distributions. Use complete sentences.

    Note:

    These answers may vary and still be correct.
  7. Describe the three most significant differences between the graphs or distributions of the theoretical and empirical distributions.

    Note:

    These answers may vary and still be correct.
  8. Does it appear that the data fit the distribution in Part 1? In 1 - 3 complete sentences, explain why or why not.
  9. Suppose that the experiment had been repeated 500 times. Which chart (from Part 2 or Part 4) would you expect to change? Why? Why wouldn’t the other chart change? How might the chart change?

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