Please see the **detailed homework policy document** for information about homework formatting, submission, and grading.

## Exercise 1

Consider a random variable \(X\) with the probability mass function

\[
f(x) = \frac{6}{3^x}, \quad x = 2, 3, 4, 5, \ldots
\]

**(a)** Find the moment-generating function of \(X\), \(M_X(t)\). Report the function, being sure to indicate the values of \(t\) where the function exists.

**(b)** Calculate \(\text{E}[X]\).

## Exercise 2

How much wood would a woodchuck chuck if a woodchuck could chuck wood? Let \(W\) denote the amount of wood a woodchuck would chuck per day (in cubic meters) if a woodchuck could chuck wood. Suppose the moment-generating function of \(W\) is

\[
M_W(t) = 0.1 \cdot e^{3t} + 0.3 \cdot e^{2t} + 0.5 \cdot e^{1t} + 0.1.
\]

**(a)** Calculate the average amount of wood a woodchuck would chuck per day, \(\text{E}[W]\).

**(b)** Calculate \(\text{Var}[W]\).

## Exercise 3

Consider a random variable \(Y\) with the probability density function

\[
f(y) = \frac{|y|}{5}, \ -1 < y < 3.
\]

**(a)** Calculate \(\text{E}[Y]\).

**(b)** Calculate \(\text{median}[Y]\), the median of \(Y\).

## Exercise 4

Suppose that scores on the previous semesterâ€™s STAT 400 Exam II were not very good. Graphed, their distribution had a shape similar to the probability density function

\[
f(s) = \frac{1}{9000}(2s + 10), \ \ 40 \leq s \leq 100.
\]

Assume that scores on this exam, \(S\), actually follow this distribution. (Note: This distribution does not necessarily reflect reality.)

**(a)** Suppose 10 students from the class are selected at random. What is the probability that (exactly) 4 of them received a score above 85?

**(b)** What was the standard deviation of the scores, \(\text{SD}[S]\)?

**(c)** What was the class 40th percentile? That is, find \(a\) such that \(P(S \leq a) = 0.40\).

## Exercise 5

Students often worry about the time it takes to complete an exam. Suppose that completion time in hours, \(T\), for the STAT 400 final exam follows a distribution with density

\[
f(t) = \frac{2}{27}(t^2+t), \ \ 0 \leq t \leq 3.
\]

What is the probability that a randomly chosen student finishes the exam during the second hour of the exam. That is, calculate \(P(1 < T < 2)\).