Final Exam, version 1
CSE 103, Fall 2013
Name:
ID:
On your desk you should have only the exam paper, writing tools, and the cheat-sheet. The cheat-sheet is one page handwritten
on both sides.
The exams are color coded. Your exam should have different colo
Final Exam, version 1
CSE 103, Fall 2013
Name:
ID:
On your desk you should have only the exam paper, writing tools, and the cheat-sheet. The cheat-sheet is one page handwritten
on both sides.
The exams are color coded. Your exam should have different colo
1) Roll two fair dice: a red die and a green die.
How many different ways are there of getting at
least one six? (all ways ways no 6) = 36-(5*5)
2) How many strings of 3 lower case English
letters are there that have the letter x in them
somewhere? Same l
Final Exam, version 2
CSE 103, Fall 2013
Name:
ID:
On your desk you should have only the exam paper, writing tools, and the cheat-sheet. The cheat-sheet is one page handwritten
on both sides.
The exams are color coded. Your exam should have different colo
Style
Size
Color
XL
L
M
S
XS
Product Set=cfw_ (
Size of Product Set=
,
, ),(
,
, ) ,.
How many different ways to arrange (all) the letters in
M I S S IS SI P P I
Consider all length 10 binary sequences.
How many contain exactly one 1?
How many contain fi
General Probability Spaces
1/10000 = 1e-4 not 9.9E-5
WebWork checks your answers against the
correct answers within some tolerance.
If you use a calculator your mistake might
be masked and reappear at a later point in the
problem.
Write complete expressio
CSE103: Introduction to
Probability and Statistics
Prof. Yoav Freund
Flipping two dice
1 dice-What is the probability that it will land on 6 ? Or on 5 ?
G,R dice. What is the probability of green=6 and red=5
R,R dice. What is the probability of red=5 a
Combinatorics
&
Uniform, finite Distributions The Combinatorial function C(n,r):
The number of ways to choose a subset
of size r from a set of size n.
Alternative notation: [ n J,
r
Expressed verally as "1] choose 1" Binomial Expansion
(a+b)2 = (,;r,+a&»)
DOCUMENT(); # This should be the first executable line in the problem.
loadMacros(
"PGstandard.pl",
"PGML.pl",
"MathObjects.pl",
"PGcourse.pl",
"parserFunction.pl"
);
TEXT(beginproblem);
$showPartialCorrectAnswers = 1;
#
Context("Numeric");
$R1=rando
Densities vs. Point Mass distributions
Mixtures
Historgrams vs. CDFs
The Kolmogorov Axioms of probability theory
lets consider the segment [0,1). Each point can be represented
by an infinite digital expansion. i.e. 0.7345231323.
To say that the set [0,1]
Burden of Proof
Legal: The burden of proof (Latin: onus probandi) is the
imperative on a party in a trial to produce the evidence
that will shift the conclusion away from the default
position to one's own position.
Example: Innocent until proven guilty.
B
2
Distribution of heights
= 5'10'
= 3'
Question: Are men telling their true height on dating sites?
Question: are men, on average, taller than women?
Question: are winners in presidential
elections taller than their opponent?
The Benford distribution
De
Exponential distribution
Normal Distribution
General way of solving problems involving
mixture of CDF
A mixture model is:
w1*P1+w2*P2+
P1,P2, are distributions
w1,w2, are the weights (or probabilities) of the components. nonnegative and sum to 1.
Find the
A Gentle introduction
to probability
The goal of this class
Probability is a branch of math.
Solving complex problems requires mathematical tools and
mathematical definitions.
It might not be obvious how the math relates to the intuition.
Some times t
Variance, Covariance,
correlation, dependence and
causation
Independent Events
Denition:
P (A B) = P (A)P (B)
What about these?
P (A B), P (A B), P (A B)
Implied by the denition
P (A B) = P (A A B) = P (A) P (A B) =
= P (A) P (A)P (B) = P (A)(1 P (B) =
A review of basic probability
and some loose ends
Independent random
variables
x, y P (X = x Y = y) = P (X = x)P (Y = y)
Consider two values for Y : y1 y2
P(X = x Y = y1 ) P(Y = y1 )
Then: x
=
P(X = x Y = y2 ) P(Y = y2 )
Similarly for any two values of X
6
P(X = i) = 2 2 ;
i
P(X = i) = 1
i=1
iP(X = i) =
i=1
Participation in this game is worth any price
(on the long term)
-6 6
Consider a game with both wins and losses
-5 5
-4
-3
3
1
-1
4
-2
2
i cfw_0,1,+1,2,+2,
1 1
1.5
P(X = i) = Z i
0
if i 0
if i = 0
Convergence to the Mean
Binomial Distribution
and
Central Limit Theorem.
Results)from)MonteVCarlo)simula:ons)
The average also called the empirical mean
Suppose X1 , X2 , Xn are
independent identically distributed (IID) random variables
Pr [ Xi = 1] = p,
CSE 103: Notes on Union Bounds, Conditional Probability and
Bayes Rule
October 30, 2014
1
Basic Probability Formulas
In lecture, we proved some basic probability formulas using Venn diagrams. Let be any outcome
space, and A and B be any two events in it.
DOCUMENT(); # This should be the first executable line in the problem.
loadMacros(
"PGstandard.pl",
"PGML.pl",
"MathObjects.pl",
"PGcourse.pl",
);
TEXT(beginproblem);
$showPartialCorrectAnswers = 1;
#
Context("Numeric");
$R1=random(8,15,1);
$R2=random
# DESCRIPTION
# Discrete mathematics, counting
# ENDDESCRIPTION
DOCUMENT(); # This should be the first executable line in the problem.
loadMacros(
"PG.pl",
"PGML.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl",
DOCUMENT(); # This should be the first executable line in the problem.
loadMacros(
"PGstandard.pl",
"PGML.pl",
"MathObjects.pl",
"PGcourse.pl",
);
TEXT(beginproblem);
$showPartialCorrectAnswers = 1;
#
Context("Numeric");
Context()->strings->add(R => c
DOCUMENT(); # This should be the first executable line in the problem.
loadMacros(
"PGstandard.pl",
"PGML.pl",
"MathObjects.pl",
"PGcourse.pl",
);
TEXT(beginproblem);
$showPartialCorrectAnswers = 1;
Context("Numeric");
$n=random(10,20,1);
$ans = Formu