SOCR EduMaterials FunctorActivities MGF Moments
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_DistributionFunctors.html .
- 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 \( E(X^2) \). 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 <math<4^{th}</math> derivatives of a MGF and set \( t=0 \)?
See also
- SOCR Home page: http://www.socr.ucla.edu
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