Philology

Problem 1 (15 pts): Choose a word with at least 5 letters and write it below:
(Fill in with YOUR OWN UNIQUE INPUT, which must be a word having 5 or more letters.
Be creative, since if others have the exact same word you will receive ZERO POINTS; for instance,
you could use your last name if it is long/unique enough, or pick an unusual word from a dictionary.)
a) (5 pts) How many di↵erent words can be written by reshu✏ing the letters above?
You must simplify your answer as an explicit integer n.
b) (5 pts) How many of these words have all vowels appearing together?
You must simplify your answer as an explicit integer n.
c) (5 pts) If you pick at random a word that can be written by reshu✏ing the letters
in your word, what is the probability that its vowels do not all appear together?
You must simplify your answer as an explicit irreducible fraction a/b.
Problem 2 (20 pts): Consider a discrete random variable X whose probability mass
function is given by:
(Fill in with YOUR OWN UNIQUE INPUT, making sure these are 3 di↵erent real numbers.
Be creative enough, since if others have the exact same numbers you will receive ZERO POINTS.)
p(x) =
8
>>>>>>>>><
>>>>>>>>>:
1
4
if x = ,
1
2
if x = ,
1
4
if x = .
a) (5 pts) Compute the probability that X takes a positive value:
You must simplify your answer as an explicit irreducible fraction a/b.
P(X > 0) =
b) (5 pts) Compute the expected value of X:
You must simplify your answer as an explicit irreducible fraction a/b.
E(X) =
c) (5 pts) Compute the expected value of X2:
You must simplify your answer as an explicit irreducible fraction a/b.
E(X2) =
d) (5 pts) Compute the variance of X:
You must simplify your answer as an explicit irreducible fraction a/b.
V ar(X) =
Problem 3 (20 pts): Suppose there are 2 bags containing the following number of
red, green, and blue marbles:
(Fill in with YOUR OWN UNIQUE INPUT, making sure these are all positive integer numbers.
Be creative enough, since if others have the exact same numbers you will receive ZERO POINTS.)
Bag A: 3
| {z } red marbles
,
| {z } green marbles
,
| {z } blue marbles
.
Bag B: 4
| {z } red marbles
,
| {z } green marbles
,
| {z } blue marbles
.
You pick 2 marbles at random from each bag, without seeing their color before picking,
and without replacing them.
a) (5 pts) What is the probability that you pick exactly 2 red marbles in total?
You must simplify your answer as an explicit irreducible fraction a/b.
b) (5 pts) If you do pick exactly 2 red marbles in total, what is the probability that
they both come from bag A?
You must simplify your answer as an explicit irreducible fraction a/b.
c) (10 pts) In general, how many of the 4 marbles you pick are expected to be red?
(Hint: Careful here, this is a somewhat lengthy computation!)
You must simplify your answer as an explicit irreducible fraction a/b.
Problem 4 (20 pts): Consider a Binomial random variable X distributed as below:
(Fill with YOUR OWN UNIQUE INPUT, making sure that n 5 is an integer, and 0 <p< 1. )
X ⇠ Binomial⇣
| {z } n
,
| {z } p
⌘
a) (5 pts) What is the expected value of X?
You must simplify your answer as an explicit irreducible fraction a/b.
E(X) =
b) (5 pts) What is the standard deviation of X?
You must simplify your answer as an explicit irreducible fraction a/b.
(X) =
c) (10 pts) What is the probability that X 2, if you know that X 1?
You must simplify your answer as an explicit irreducible fraction a/b.
P(X 2 | X 1) =
Problem 5 (15 pts): Let X and Y be as in Problem 1 of Homework 9.
a) (5 pts) Are X and Y independent? Justify.
b) (5 pts) Compute the expected value of Y .
E(Y ) =
c) (5 pts) Compute the expected value of XY .
E(XY ) =
Problem 6 (10 pts): According to a study, average text messages currently exchanged
by users in the USA contain approximately 7 words and 1 emoji. Assume that the
number of words and emojis in a text message are independent from one another.
a) (5 pts) Use Markov’s inequality to estimate from above the probability that a text
message contains at least 10 words and 2 emojis.
You must simplify your answer as an irreducible fraction a/b.
b) (5 pts) Suppose that the standard deviation of the number of words in a text message is 2. Use Chebyshev’s inequality to estimate from below the proportion
of text messages that contain between 4 and 10 words.
You must simplify your answer as an irreducible fraction a/b.