## Transcribed Text

MATH
1. Identify whether the following statements are true or false. If the statement
is false then correct it.
(a.) The number of ways to sample k items from n objects with replacement
and no regards to order is denoted Ck,n.
(b.) Let X and Y be joint random variables with joint PDF, fpx, yq. Let
f1pxq be the marginal of X and f2pyq be the marginal of Y . Then gpy|xq “ fpx,yq
f2pyq
.
(c.) Let n and k be integers with n ě k. then
Pk,n “ npn ´ 1qpn ´ 2q. . .pn ´ k ´ 1q.
(d.) The probability mass function, ppxq, for an experiment gives the probability for each element x in the sample space.
(2.) Identify whether the following statements are true or false:
(a.) Let S be any sample space and A be an event in S. Then PpAq “ |A|
|S|
.
(b.) Two events A and B are independent if and only if they are mutually
exclusive.
(c.) Random variables X and Y are joint random variables if and only if
there exists function f with fpx, yq “ fXpxqfY pyq for some functions fXpxq and
fY pyq.
(d.) The collection of events A, B, and C are independent if and only if
PpA X B X Cq “ PpAqPpBqPpCq.
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(3.) Twelve individuals want to form a committee of four.
(a.) How many committees are possible?
(b.) The 12 individuals consist of 5 biologists, 4 chemists, and 3 physicists. How many
committees consisting of 1 biologist, 1 chemist, and 2 physicists are possible?
(c.) In the setting of part (b), if all committees are equally as likely, what is the
probability the committee formed will consist of 1 biologist, 1 chemist, and 2
physicists?
(4.) Let the random variables X and Y have the joint PDF
fpx, yq “ kxy2
for 0 ď x ď 2, x ď y ď 3.
(a.) Find the constant k.
(b.) Find the joint CDF of X and Y .
(5.) You are out of town at a research conference; in your suitcase you have two white
shirts, 3 black shirts, 2 pairs of black slacks, and 3 pairs of gray slacks. What is the
likelihood you wear an all black outfit?
(6.) In many states the license plates consist of a string of seven characters such that the
first three are letters and the second four are numbers. Assume each string of characters
are equally as likely and repetition is allowed on a single license plate.
(a.) Describe the random variable, X, that counts the number of vowels in each license
plate. (Do not consider y as a vowel.)
(b.) Give the probability mass function, ppxq, for this scenario.
(c.) Use it to find the likelihood a license plate will have three vowels.
(7.) Let
X “ t35, 45, 39, 41, 41, 44, 46, 48, 49, 34, 12, 50, 20, 38, 40u
(a.) Insert X into R.
(a.) Find the mean of X.
(b.) Depict X in a boxplot.
(NOTE: This question should be answered entirely using code for R.)
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(8.) Let X have PMF
x 1 2 3 4
ppxq 0.4 0.3 0.1 0.2
(a.) Calculate V arpXq and V arp1{Xq.
(b.) In a win-win game, the player will win a monetary prize, but has to decide between
the fixed prize of $1000/EpXq and the random price of $1000/X, where the
random variable X has the PMF given above. Which choice would you recommend
the player make?
(9.) Consider the bivariate density function
fpx, yq “ 12
7
px
2 ` xyq for 0 ď x ď 1 and 0 ď y ď 1.
Find the regression function of X on Y . Use it to find the expected value of X given
Y “ 0.5.
(10.) Two students, A and B, are both registered for a certain course. Assume that student
A attends class 80 percent of the time and student B attends class 60 percent of the
time, and the absences of the two students are independent.
(a.) What is the probability that at least two of the students will be in class on a
given day?
(b.) It at least one of the two students is in class on a given day, what is the probability
that A is in class that day?

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