Sam Tripp
ST372
Homework0 22AUG
Due 29AUG
ST 372 Webpage: http://www4.stat.ncsu.edu/~bmasmith/ST372F13/
HUMUHUMUNUKUNUKUAPUAA
How many ways can you arrange “Humuhumunukunukuapuaa?”
Consider arranging “Dog.” How many significant ways can you rearrange the letters to make a new,
distinct word?
DOG DGO ODG OGD GDO GOD = 6
So, what if we did “DoD?” Shouldn’t it be the same number of permutations?
DOD DDO ODD = 3
How is this? Notice two of the letters are the same.
Consider only the distinct letters, because swapping the same letter in each respective space would
yield the same word, right?
So, we must consider the multiplicities of letters in words.
D O G – Three different letters. One of each. The number of permutations you’ll get should be the
factorial of 3.
3! = 3x2x1 = 6
D O D – Two different letters. One “O” and two “Ds.” So, should this be the factorial of 2?
2! = 2x1
If you were to assume that each letter were different, you’d get 6 permutations in a three letter word,
but with two of the letters, you’d have to divide that number by the product of the factorials for each
multiplicity of each letter.
For instance, DOG would be 3! Divided by 1! For D, 1! For O, and 1! For G.
3!/(1!x1!x1!) = 3x2x1 / 1x1x1 = 6/1 = 6
Let’s try DOD. DOD would be 3! Divided by 2! For D and 1! For O.
3!/(2!x1!) = 6/2 = 3.
Apply this to “the fish with the pig-like nose.”
Humuhumunukunukuapuaa has a 21 total individual letters.
The multiplicities: 2! For H 9! For U 2! For M 2! For N 2! For K 3! For A and 1! For P
So, we should have 21!/(2!x9!x2!x2!x2!x3!x1!) which should yield 1.4666x10^12.
Mahalo!
TOOTSIE ROLLS (Combinations, i.e. “n” choose “k”)
At the beginning of class, 41 students randomly pulled 2 tootsie rolls from a fiveflavored assorted bag. Five of those 41 students pulled two of the same flavored
tootsie rolls. What are the chances of this occurring?
Think of Combinations. What is a combination? A combination is a way of selecting
many things from a larger group. In this case, we are selecting two identical tootsie
rolls out of a large assortment of tootsie rolls.
Lets think of the five different flavors first. Lets say the bag only consisted of five
distinct tootsie rolls (Apple, Orange, Lime, Banana and Grape), and you were to
choose 2 of them randomly. How many different possible combinations of 2 distinct
flavors could you choose from?
AO AL AB AG OL OB OG LB LG BG = 10
Now, how many different possible combinations of 2 distinct flavors could you
choose from a whole bag of them?
Don’t Recall the Endgame.
Dr. Pepper
Professor Smith brought out two displays of empty and distinct Dr. Pepper cans, as
well as other brands that led with “Dr.”. He was able to gather quite a bit of data
from an open house he went to, in which he determined who enjoyed the drinks that
he had in the display.
Naturally, and unbiased, the data he collected was based on individual taste and
didn’t have any correlations with anything else (random). Plotting this data on a bar
graph, the variance was distinct and obvious.
What would cause this graph to become linear?
Two Options on Grading
We were given two options on grading. Unfortunately, I cannot remember both, but
the one chosen was the best option by far. In general, it was to drop the lowest of
three quizzes and the final exam.
Linear Combinations (Chapter 5 Section 5)
A handout was… handed out involving Linear Combinations of Random Variables
and Approximation for Mean and Variance.
In general
1. The sum of two independent normally distributed random variables is
normal, with its mean being the sum of the two means, and its
variance being the sum of the two variances (i.e., the square of the
standard deviation is the sum of the squares of the standard
deviations).
2. You need to know how to find the mean or expected value of a linear
combination (CH5 Section 5 Proposition 1)
3. You need to know how to find the variance of a linear combination
(Proposition 2)
4. Of a Normal or Gaussian Random Variable, f(x;mu, sigma)
a. How to go from X to Z by subtracting mean and dividing
difference by standard deviation.
b. How to compute probabilities of events using the cumulative
normal table in front of text (Table A.3 of Devore, 8thEd)
5. Using Taylor Series Formula for approximate mean and variance
(Find this on Wikipedia at
http://en.wikipedia.org/wiki/Taylor_expansions_for_the_moments_of
_functions_of_random_variables) (Take note of the final expected
value of the First Moment where mu sub x and sigma squared sub x
are the mean and variance of X respectively.)
Problem 93 of Chapter 5 in Devore, 8thEd
Do it.