11.3.2.2. Derivatives

Try these first

1. 1.

   What is the definition of a moment generating function?

   Hint

   Think about how moments relate to expected values.

2. 2.

   What does the first derivative of an MGF evaluated at zero represent?

   Hint

   Consider what the mean describes in a distribution.

3. 3.

   What is the main use of a moment generating function?

   • To define probabilities
   • To summarize moments
   • To calculate means only

   Hint

   Think about the summary aspect of probability distributions.

4. 4.

   True or False: The second moment of a distribution is always the variance.

   • True
   • False

   Hint

   Remember the formula for variance.

5. 5.

   Given a random variable with MGF M(t) = (1 - 2t)^-3, find the mean and variance.

   Hint

   Use the properties of the derivatives of MGFs to find moments.

6. 6.

   Design an MGF for a random variable that has a mean of 5 and variance of 2.

   Hint

   Consider how you can use the mean and variance to shape the MGF.