Moment generating functions

2 free worked examples

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1. Exam PMoment generating functionsCore

   A random variable has $M(t) = e^{3t + 8t^{2}}$. Find its variance.

   1. A$3.0000$
   2. B$8.0000$
   3. C$9.0000$
   4. D$16.0000$
   5. E$64.0000$

   Solution

   1. The normal MGF is $\exp\left(\mu t + \dfrac{\sigma^{2}t^{2}}{2}\right)$, so match coefficients.
   2. The $t$ coefficient gives $\mu = 3$.
   3. The $t^{2}$ coefficient gives $\dfrac{\sigma^{2}}{2} = 8$, so $\sigma^{2} = 16$.
   4. The variance is 16. Recognising the shape is far faster than differentiating twice.

   Trap. Reading the t² coefficient as the variance and answering 8.

2. Exam PMoment generating functionsExam level

   A variable has $M(t) = 0.2 + 0.3e^{t} + 0.5e^{3t}$. Find $E[X]$.

   1. A$0.8000$
   2. B$1.5000$
   3. C$1.8000$
   4. D$2.0000$
   5. E$3.0000$

   Solution

   1. This MGF is a weighted sum of $e^{tx}$ terms, so it describes a discrete variable taking values 0, 1 and 3 with probabilities 0.2, 0.3 and 0.5.
   2. Read the distribution straight off the exponents and coefficients.
   3. $E[X] = 0(0.2) + 1(0.3) + 3(0.5)$.
   4. $= 1.8$. Differentiating and setting $t = 0$ gives the same, but reading it off is faster and less error-prone.

   Trap. Differentiating carelessly and dropping the factor of 3 from the e^{3t} term.