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Consider a random variable X with the probability mass function
f(x)=63x,x=2,3,4,5,…
(a) Find the moment-generating function of X, MX(t). Report the function, being sure to indicate the values of t where the function exists.
(b) Calculate 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
MW(t)=0.1⋅e3t+0.3⋅e2t+0.5⋅e1t+0.1.
(a) Calculate the average amount of wood a woodchuck would chuck per day, E[W].
(b) Calculate Var[W].
Consider a random variable Y with the probability density function
f(y)=|y|5, −1<y<3.
(a) Calculate E[Y].
(b) Calculate 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)=19000(2s+10), 40≤s≤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, SD[S]?
(c) What was the class 40th percentile? That is, find a such that P(S≤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)=227(t2+t), 0≤t≤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).