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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 \(\text{E}[X]\).
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 \(\text{Var}[W]\).
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 \(Y\).
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 \(P(S \leq a) = 0.40\).
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 \(P(1 < T < 2)\).