11.3.2. Properties of MGFs

Try these first

1. 1.

   What is the formula for the moment generating function of a random variable X?

   Hint

   Think about expectations and exponentials.

2. 2.

   True or False: If two random variables are independent, their MGFs can be added.

   Hint

   Focus on the properties of MGFs.

3. 3.

   What does the moment generating function M_X(t) represent?

   • It calculates probabilities
   • It summarizes moments
   • It generates random variables

   Hint

   Think about what MGFs do in relation to random variables.

4. 4.

   True or False: The first moment can be derived from the first derivative of the MGF.

   • True
   • False

   Hint

   Remember how derivatives relate to moments.

5. 5.

   A discrete random variable has an MGF of M_X(t) = (0.5 * e^t + 0.5 * e^2t). Calculate the first two moments using this MGF.

   Hint

   Differentiate MGF twice, evaluate at t=0.

6. 6.

   Given two independent random variables with MGFs M_X(t) = e^(2t) and M_Y(t) = e^(3t), find M_{X+Y}(t).

   Hint

   Use the property of additivity of MGFs.