11.3. Moment Generating Functions (MGFs)

Try these first

1. 1.

   What is the definition of a Moment Generating Function?

   Hint

   Think about the expected value.

2. 2.

   List the property that allows obtaining the r-th moment from an MGF.

   Hint

   Consider how derivatives work.

3. 3.

   What is the calculation for the first moment using an MGF?

   • M_X(0)
   • M'_X(0)
   • E[X^2]

   Hint

   Consider what derivation reveals about the function.

4. 4.

   True or False: MGFs can be used for dependent random variables.

   • True
   • False

   Hint

   Reflect on how independence affects calculations.

5. 5.

   If X ~ N(0,1) has an MGF given by M_X(t) = exp(t^2/2), derive E[X^2] and variance.

   Hint

   Differentiate the MGF to find moments.

6. 6.

   Consider a random variable with probabilities P(X=1)=1/3, P(X=2)=2/3. Find the MGF, first moment, and variance.

   Hint

   Make sure to differentiate and evaluate properly.