DS-GA 1002 Lecture notes 1
September 14, 2015
Probability
1
Introduction
Probability theory is a tool that allows us to reason mathematically about uncertainty. In
this notes we will learn how to build probabilistic models to incorporate the information w

Statistical and Mathematical Methods DS-GA 1002
December 14, 2015
Final
The duration of the exam is 3 hours.
The exam is open book. You may use electronic devices but only to view the lecture
notes. No internet is allowed. Calculators are permitted.
Yo

Statistical and Mathematical Methods DS-GA 1002
November 5, 2015
Homework 6 Solutions
1. (10 points) Uniform distribution.
a. u has to be larger than x(n) = maxcfw_x1 , x2 , . . . , xn .
b. The likelihood function is
Lx (u) =
n
Y
fXi (xi , u)
(1)
1/u = 1/

Statistical and Mathematical Methods DS-GA 1002
November 17, 2015
Homework 8 Solutions
1. (10 points) Probability of error vs MSE (10 points).
a. We can easily find the piecewise constant density of Y
1
4 |y| 1
fY (y) =
1
1 < |y| 3
8
0
otherwise
The cond

Statistical and Mathematical Methods DS-GA 1002
November 11, 2015
Homework 7 Solutions
1. (10 points) Short questions.
a. No. In fact, it can still be quite clear that it doesnt hold, but we cannot rule it out up to
the significance level that we want to.

Statistical and Mathematical Methods DS-GA 1002
October 19, 2016
Homework 5
Due Friday, October 28
Submit your assignment through NYU classes in pdf format.
1. Basketball. A data analyst working for the Golden State Warriors determines that Stephen
Curry

DS-GA 1002 Lecture notes 7
November 9, 2015
Statistics: Estimation
Estimation is a classical problem in statistics. We have access to samples from a random
vector Y and want to estimate the corresponding samples of a related random variable X
(X could als

DS-GA 1002 Lecture notes 2
September 21, 2015
Probability (continued)
1
Random variables (continued)
1.1
Conditioning on an event
Given a random variable X with a certain distribution, imagine that it is revealed that
X S, where S R. In order to incorpora

Statistical and Mathematical Methods DS-GA 1002
December 9, 2015
Homework 11 Solutions
1. (10 points) Convexity (10 points).
a. For any 0 1 and any x, y R,
f (x + (1 ) y) =
n
X
i=1
n
X
ai fi (x + (1 ) y)
(1)
ai (fi (x) + (1 ) fi (y)
i=1
n
X
=
ai fi (x) +

Statistical and Mathematical Methods DS-GA 1002
November 2, 2016
Homework 7
Due Friday, November 11
Submit your assignment through NYU classes in pdf format.
1. Rejection sampling for discrete data. In this problem we will discuss a rejection sampling met

Statistical and Mathematical Methods DS-GA 1002
November 9, 2016
Homework 8
Due Friday, November 18
Submit your assignment through NYU classes in pdf format.
1. Descriptive statistics. Find a dataset online that has between 5 and 10 features per example.

Statistical and Mathematical Methods DS-GA 1002
October 26, 2016
Homework 6
Due Friday, November 4
Submit your assignment through NYU classes in pdf format.
1. Radioactive sample. Consider the following experiment. We have a radioactive sample situated
at

Statistical and Mathematical Methods DS-GA 1002
November 25, 2016
Homework 10
Due Friday, December 2
Submit your assignment through NYU classes in pdf format.
1. Sign test. Your friend is convinced that in general the left ear of most people is longer tha

Statistical and Mathematical Methods DS-GA 1002
September 14, 2016
Homework 1
Due Wednesday, September 21
Submit your assignment through NYU classes.
1. True or False. Prove the following statements or provide a counterexample. Let A and B be
two events i

Statistical and Mathematical Methods DS-GA 1002
November 16, 2016
Homework 9
Due Sunday, November 27
Submit your assignment through NYU classes in pdf format.
1. Empirical probability mass function. You are hired to estimate the distribution of ages (in y

Statistical and Mathematical Methods DS-GA 1002
December 16, 2015
Final Solutions
1. Short questions (20 points)
a. Any vector in span cfw_u1 , u2 can be written as x := u1 + u2 for some , R. By
linearity
Ax = Au1 + Au2
= u1 + u2
= x,
(1)
(2)
(3)
where i

DS-GA 1002 Lecture notes 5
October 19, 2015
Statistics: Learning models from data
Learning models from data that are assumed to be generated probabilistically from a certain
unknown distribution is a crucial step in statistical analysis. The model may be

DS-GA 1002 Lecture notes 0
September 14, 2015
Set theory
1
Basic definitions
A set is a collection of objects. The set of all elements that we consider in a certain situation
is called the universe and is usually denoted by . If an object x in belongs to

DS-GA 1002 Lecture notes 4
October 2, 2015
Statistics
1
Estimation of population parameters
The goal of statistics is to extract information from data and evaluate the uncertainty of
this information quantitatively. In this section we consider the problem

DS-GA 1002 Lecture notes 10
November 30, 2015
Optimization
1
Motivation
In data analysis, optimization methods are used to fit models to the available data. Model
parameters are chosen by maximizing or minimizing a cost function, such as the likelihood of

DS-GA 1002 Lecture notes 8
November 9, 2015
Linear models: Algebra
Linear models are a pillar of modern data analysis. Many phenomena can be modeled as
linear, at least approximately. In addition, linear models tend to be easy to interpret (quantity A is

DS-GA 1002 Lecture notes 9
November 16, 2015
Linear models
1
Projections
The projection of a vector x onto a subspace S is the vector in S that is closest to x. In
order to define this rigorously, we start by introducing the concept of direct sum. If two

DS-GA 1002 Lecture notes 10
November 23, 2015
Linear models
1
Linear functions
A linear model encodes the assumption that two quantities are linearly related. Mathematically, this is characterized using linear functions. A linear function is a function su

Statistical and Mathematical Methods DS-GA 1002
October 27, 2015
Homework 6
Due Tuesday, November 3
Please either give the assignment to Loraine at the CDS or send it via email
to the graders before noon.
1. Uniform distribution. You have data that you ha

Statistical and Mathematical Methods DS-GA 1002
December 8, 2015
Sample Final Problems Solutions
1. Short questions
a. Ax = b has a solution if b is in the range of A. The dimension of the range of A is n because
A has n linearly-independent columns. Sinc

Statistical and Mathematical Methods DS-GA 1002
December 6, 2015
Sample Final Problems
1. Short questions
a. For a m n matrix A with rank n, where m > n, is it possible for Ax = b to have no
solution?
b. For a full-rank m n matrix A with rank m, where n >

Statistical and Mathematical Methods DS-GA 1002
November 25, 2015
Homework 9 Solutions
1. (15 points) Projections.
a. False. This only holds if the vectors in the basis are orthogonal. Take
0
1
b=
, b2 =
.
1
1
(1)
This is obviously a basis of R2 . How

Statistical and Mathematical Methods DS-GA 1002
November 23, 2015
Homework 10
Due Tuesday, December 8
Please either give the assignment to Loraine at the CDS or send it via email
to the graders before noon.
1. Statements (10 points). Prove the following s

Statistical and Mathematical Methods DS-GA 1002
November 16, 2015
Homework 9
Due Tuesday, November 24
Please either give the assignment to Loraine at the CDS or send it via email
to the graders before noon.
1. True or false (10 points). Are these statemen

Statistical and Mathematical Methods DS-GA 1002
November 11, 2015
Homework 8
Due Tuesday, November 17
Please either give the assignment to Loraine at the CDS or send it via email
to the graders before noon.
1. Probability of error vs MSE (10 points). We a