## 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 semester’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 $$P(1 < T < 2)$$.