SOCR EduMaterials Activities Exponential Distribution

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(→This is an activity to explore the Exponential Distribution: Learn how to compute probabilities, densities, and percentiles.)
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* '''Description''':  You can access the applet for the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html Exponential Distributions]  
* '''Description''':  You can access the applet for the [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html Exponential Distributions]  
 +
* Here is the shape of the exponential distribution (this is a snpashot from the SOCR website:
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\begin{itemize}
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<center>[[Image:SOCR_Activities_DieCoinExperiment_Christou_092206_Fig1.jpg |600px]]</center>
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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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* '''Exercise 1:''' Graph and print:
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$X \sim exp(10)$
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a. exp(0.2)
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\begin{itemize}
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b. exp(1)
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\item[1.] Locate the maximum density for each one of these distributions.   
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c. exp(10)
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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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* '''Exercise 2:''' Locate the maximum density for each one of these distributions.   
 +
 
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* '''Exercise 3:''' 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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* '''Exercise 4:''' Find one percentile for each of these distributions and record them on the printouts.  Verify these  percentiles using the formula we discussed in class:
\[
\[
x_p=\frac{ln(1-\frac{p}{100})}{-\lambda}
x_p=\frac{ln(1-\frac{p}{100})}{-\lambda}
\]
\]
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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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* '''Exercise 5:''' Compute one cumulative probability for each one of these distributions, show it on the graph, and verify it with the formula:
\[
\[
P(X \le x)=1-e^{-\lambda x}
P(X \le x)=1-e^{-\lambda x}
\]
\]
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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}
 
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* '''Exercise 1''': Construct the joint probability distribution of X and Y.
 
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* '''Exercise 2''': Find the conditional expected value of Y given X=5.
 
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* '''Exercise 3''': Find the conditional variance of Y given X=5.
 
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* '''Exercise 4''':  Find the expected value of Y.
 
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* '''Exercise 5''': Find the standard deviation of Y.
 
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* '''Exercise 6''': Graph the probability distribution of Y.
 
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* '''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:
 
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a. n = 1000 times.
 
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b. n= 10000 times
 
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<center>[[Image:SOCR_Activities_DieCoinExperiment_Christou_092206_Fig1.jpg |600px]]</center>
 
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* '''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.
 
<hr>
<hr>

Revision as of 00:57, 24 September 2006

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

  • Here is the shape of the exponential distribution (this is a snpashot from the SOCR website:


  • Exercise 1: Graph and print:

a. exp(0.2) b. exp(1) c. exp(10)

  • Exercise 2: Locate the maximum density for each one of these distributions.
  • Exercise 3: 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).
  • Exercise 4: Find one percentile for each of these distributions and record them on the printouts. Verify these percentiles using the formula we discussed in class:

\[ x_p=\frac{ln(1-\frac{p}{100})}{-\lambda} \]

  • Exercise 5: Compute one cumulative probability for each one of these distributions, show it on the graph, and verify it with the formula:

\[ P(X \le x)=1-e^{-\lambda x} \]


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