Processing math: 100%

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)=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].

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

MW(t)=0.1e3t+0.3e2t+0.5e1t+0.1.

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

(b) Calculate Var[W].

Exercise 3

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.

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)=19000(2s+10),  40s100.

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(Sa)=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)=227(t2+t),  0t3.

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).