11.3.1. Definition

Try these first

1. 1.

   Define a Moment Generating Function.

   Hint

   Think about how functions can capture properties of distributions.

2. 2.

   What is the first moment obtained from an MGF?

   Hint

   Consider the relationship to the expectation operator.

3. 3.

   What is the MGF of a random variable X?

   • E[X]
   • E[e^{tX}]
   • E[X^2]
   • E[X+Y]

   Hint

   Remember the functional form used for MGFs.

4. 4.

   True or False: The MGF uniquely determines the distribution of a random variable.

   • True
   • False

   Hint

   Think about the implications of having the same MGF.

5. 5.

   A random variable X has an MGF M_X(t) = e^{(3 + t)/2}. Find the mean and variance of X.

   Hint

   Start with derivative calculations for moments.

6. 6.

   For two independent random variables X and Y with known MGFs, compute the MGF for Z = X + Y.

   Hint

   Don't forget the independence property while calculating.