SOCR EduMaterials Activities ExpDist

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You can access the applet for the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html Exponential Distributions]  
You can access the applet for the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html Exponential Distributions]  
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* Graph and print
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**<math>X \sim exp(0.2)</math><br>
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**<math>X \sim exp(1)</math> <br>
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**<math>X \sim exp(10)</math>
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<math>\begin{itemize}
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* Locate the maximum density for each one of these distributions.   
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\item[a.] Graph and print \\
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$X \sim exp(0.2)$ \\
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$X \sim exp(1)$ \\
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$X \sim exp(10)$
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\begin{itemize}
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\item[1.] Locate the maximum density for each one of these distributions.   
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\item[2.] Find the height of the density at 3 values of $X$ (one near 0, one near the mean, and one towards the tail of the distribution).
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\item[3.] Find one percentile for each of these distributions and record them on the printouts.  Verify these  percentiles using the formula we discussed in class:
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\[
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x_p=\frac{ln(1-\frac{p}{100})}{-\lambda}
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\]
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\item[4.] Compute one cumulative probability for each one of these distributions, show it on the graph, and verify it with the formula:
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\[
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P(X \le x)=1-e^{-\lambda x}
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\]
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\end{itemize}
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\item[b.] Graph and print \\
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$X \sim N(2,0.5)$ \\
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$X \sim N(10,2)$ \\
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$X \sim N(20,5$
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\begin{itemize}
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\item[1.] Find one percentile for each one of these distributions and locate them on the printouts.
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\item[2.] Find one cumulative probability for each one of these distributions and locate them on the printouts.
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\end{itemize}</math>
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* Find the height of the density at 3 values of <math>X</math> (one near 0, one near the mean, and one towards the tail of the distribution).
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* Find one percentile for each of these distributions and record them on the printouts.  Verify these  percentiles using the formula we discussed in class:
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<math>\[ x_p=\frac{ln(1-\frac{p}{100})}{-\lambda}\]</math>
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* Compute one cumulative probability for each one of these distributions, show it on the graph, and verify it with the formula:
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<math>\[ P(X \le x)=1-e^{-\lambda x} \]</math>
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* Graph and print
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** <math>X \sim N(2,0.5)</math>
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** <math>X \sim N(10,2)</math>
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** <math>X \sim N(20,5)</math>
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* Find one percentile for each one of these distributions and locate them on the printouts.
 +
* Find one cumulative probability for each one of these distributions and locate them on the printouts.
=='''Exercise 1'''==
=='''Exercise 1'''==
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=='''Exercise 7'''==
=='''Exercise 7'''==
Use SOCR [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html Experiments] and choose "Die Coin Experiment" to graph and print the empirical distribution of Y when the experiment is performed:
Use SOCR [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html Experiments] and choose "Die Coin Experiment" to graph and print the empirical distribution of Y when the experiment is performed:
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* n = 1000 times.
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a. n = 1000 times.
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* n= 10000 times
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b. n= 10000 times
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<center>[[Image:SOCR_Activities_DieCoinExperiment_Christou_092206_Fig1.jpg |600px]]</center>
<center>[[Image:SOCR_Activities_DieCoinExperiment_Christou_092206_Fig1.jpg |600px]]</center>

Revision as of 17:14, 12 June 2007

Contents

This is an activity to explore the Exponential Distribution: Learn how to compute probabilities, densities, and percentiles

Description

You can access the applet for the Exponential Distributions

  • Graph and print
    • X \sim exp(0.2)
    • X \sim exp(1)
    • X \sim exp(10)
  • Locate the maximum density for each one of these distributions.
  • Find the height of the density at 3 values of X (one near 0, one near the mean, and one towards the tail of the distribution).
  • Find one percentile for each of these distributions and record them on the printouts. Verify these percentiles using the formula we discussed in class:

Failed to parse (lexing error): \[ x_p=\frac{ln(1-\frac{p}{100})}{-\lambda}\]


  • Compute one cumulative probability for each one of these distributions, show it on the graph, and verify it with the formula:

Failed to parse (lexing error): \[ P(X \le x)=1-e^{-\lambda x} \]


  • Graph and print
    • X \sim N(2,0.5)
    • X \sim N(10,2)
    • X \sim N(20,5)
  • Find one percentile for each one of these distributions and locate them on the printouts.
  • Find one cumulative probability for each one of these distributions and locate them on the printouts.

Exercise 1

Construct the joint probability distribution of X and Y.


Exercise 2

Find the conditional expected value of Y given X=5.


Exercise 3

Find the conditional variance of Y given X=5.


Exercise 4

Find the expected value of Y.


Exercise 5

Find the standard deviation of Y.


Exercise 6

Graph the probability distribution of Y.


Exercise 7

Use SOCR Experiments and choose "Die Coin Experiment" to graph and print the empirical distribution of Y when the experiment is performed:

  • n = 1000 times.
  • n= 10000 times

Exercise 8

Compare the theoretical mean and standard deviation of Y (exercise 4 and 5) with the empirical mean and standard deviation found in exercise 7.

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Retrieved from "http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_ExpDist"
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