SOCR EduMaterials FunctorActivities MGF Moments

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(New page: == This is an activity to explore useful properties of MGF's.== * '''Description''': You can access the applets for the above distributions at http://www.socr.ucla.edu/htmls/SOCR_Distri...)
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* '''Description''':  You can access the applets for the above distributions at  http://www.socr.ucla.edu/htmls/SOCR_DistributionFunctors.html .
* '''Description''':  You can access the applets for the above distributions at  http://www.socr.ucla.edu/htmls/SOCR_DistributionFunctors.html .
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* '''Exercise 1:''' As you have learned in class, there are quite a few interesting properties that Moment Generating Functions hold.  For example you learned that <math> E(X^n)=M_{x}^{(n)}(0)={d^n M_x(t)\over{dt^n}}\mid_{t=0} </math> If the MGF is defined in the neighborhood of 0.  So to get the Expected Value for a particular distribution, you would take the first derivative of the MGF and set t=0.  Use SOCR to graph and print the following distributions and answer the questions below.  ''You must do these exercises using MGF's, you can find the slope using the mouse pointer.''
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**a.  Find the Expected Value of <math> X \sim Binomial(10,.5) </math>
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**b.  Find the Expected Value of <math> X \sim Normal(0,1) </math>
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**c.  Find the Expected Value of <math> X \sim ChiSquare(13) </math>
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* '''Exercise 2:''' Can you use MGF's to find the Expected Value for the Continuous Uniform Distribution?  Why or why not?
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* '''Exercise 3:'''  In Exercise 1, we calculated the 1^{st} Moment.  If we take the second derivative of the MGF with respect to t, where <math> t=0 </math>.  We get <math> E(X^2) </math>.  We can use this to find the Variance of a particular Distribution.  Repeat Parts (a,b,c) for Exercise 1, but this time calculate the variance. 
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* '''Exercise 4:'''  What do we get when we take the 3^{rd} and 4^{th} derivatives of a MGF and set <math> t=0 </math>?

Revision as of 03:22, 9 January 2008

This is an activity to explore useful properties of MGF's.

  • Exercise 1: As you have learned in class, there are quite a few interesting properties that Moment Generating Functions hold. For example you learned that E(X^n)=M_{x}^{(n)}(0)={d^n M_x(t)\over{dt^n}}\mid_{t=0} If the MGF is defined in the neighborhood of 0. So to get the Expected Value for a particular distribution, you would take the first derivative of the MGF and set t=0. Use SOCR to graph and print the following distributions and answer the questions below. You must do these exercises using MGF's, you can find the slope using the mouse pointer.
    • a. Find the Expected Value of X \sim Binomial(10,.5)
    • b. Find the Expected Value of X \sim Normal(0,1)
    • c. Find the Expected Value of X \sim ChiSquare(13)
  • Exercise 2: Can you use MGF's to find the Expected Value for the Continuous Uniform Distribution? Why or why not?
  • Exercise 3: In Exercise 1, we calculated the 1^{st} Moment. If we take the second derivative of the MGF with respect to t, where t = 0. We get E(X2). We can use this to find the Variance of a particular Distribution. Repeat Parts (a,b,c) for Exercise 1, but this time calculate the variance.
  • Exercise 4: What do we get when we take the 3^{rd} and 4^{th} derivatives of a MGF and set t = 0?
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