11.5. Examples

Try these first

1. 1.

   What is the MGF of a discrete random variable taking values 0 and 1 with equal probability?

   Hint

   Use the definition of MGF, E[e^(tX)].

2. 2.

   Calculate the mean of a discrete random variable that takes on values 1 and 3 with probabilities 1/4 and 3/4 respectively.

   Hint

   Use the formula E[X] = Σ(x * P(X=x)).

3. 3.

   What is the MGF of a discrete random variable with possible values 0 and 1 each with probability 1/2?

   • (1 + e^t)/2
   • (1 + e^(t/2))/2
   • e^t

   Hint

   Recall the formula for the MGF.

4. 4.

   True or False: The mean of a normally distributed random variable is always zero.

   • True
   • False

   Hint

   Consider the definition of a normal distribution.

5. 5.

   A discrete random variable X takes values 2, 3, and 5 with probabilities 0.3, 0.5, and 0.2 respectively. Calculate the MGF and derive the mean and variance.

   Hint

   Use the definition of MGF and the formulas for mean and variance.

6. 6.

   Assume Y ~ N(μ, σ²). Prove that the mean can be derived from the MGF.

   Hint

   Differentiate the MGF and evaluate at t=0.