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Discrete Random Variables: Lab I

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.

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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.
 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.
3. Build the theoretical PDF chart for X based on the distribution in Part I.
 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.
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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