STDS.2026.Week1

1.

Statistical Techniques for Data
Science Week 1

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Objectives (for today)
● to learn about the course
● to recall Probability theory
● to do short introduction to Statistics
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Course. Team and Communication
● Zamira Kholmatova (@zzzkh)
● Okibe Solomon Okibe
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4.

Course. Syllabus
● Statistics
● Non-parametric Statistics
● Bandit algorithms
● Sampling
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5.

Course. Structure
● Lectures (2 hours/week) + Quizzes (5-10 min. on a lecture)
● Labs (2 hours/week) + Assignments (after each lab)
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6.

Course. Books
● Hogg, Robert V., Joseph W. McKean, and Allen T. Craig. «Introduction to
Mathematical Statistics»
● Hogg, Robert V., Elliot A. Tanis, and Dale L. Zimmerman. «Probability and
statistical inference»
● Hastie, T. Tibshirani, R. and Friedman, J. «The Elements of Statistical Learning»
● *Bishop, C. «Pattern Recognition and Machine Learning»
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7.

Course. Assessment
● Quizzes:
● Labs (homeworks):
● Assignments:
● Midterm:
● Final Exam:
10 %
10 %
30 %
85-100 A
70-84 B
55-69 C
0-54
D
25 %
25 %
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8.

Course. Tools
● Pen and Paper
● Python (ver. 3+)
● …
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9.

Prerequisites
● Linear Algebra
● Calculus
● Probability Theory
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How to success?
● Assignments:
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work hard (individually) + office hours
● Exams:
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read the books + office hours
do exercises
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11.

What are your expectations?
● Answer the questions:
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Which book I will read during the course?
What is the most interesting project topic for me?
Which tools I would like to study in the course?
Which paper(s) I would like to read / implement ?
● Fill the form: https://forms.yandex.ru/cloud/696be4e0f47e734bf2544df2
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Part1. Review of Probability
Theory

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Warm up… Random Variables
● What are examples of random variables (YES/NO)?
A. Winning a lottery
B. Choosing a green ball from an urn with a large mixture of red and black balls
C. Total value from a roll of two dice
D. Two people in a classroom sharing the same birthday
E. The average exam score of a class if every student guesses answers
F. Winings from a game with a $1 gain/loss for each head/tail coin flip in a series of 10 flips
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14.

Random Variables
● What are examples of random variables?
A. Winning a lottery
B. Choosing a green ball from an urn with a large mixture of red and black balls
C. Total value from a roll of two dice
D. Two people in a classroom sharing the same birthday
E. The average exam score of a class if every student guesses answers
F. Winings from a game with a $1 gain/loss for each head/tail coin flip in a series of 10 flips
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Terms
● Do not confuse/mix
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Outcomes,
Sample Space,
Events and
Random Variables
● A random variable assigns a numerical value to each outcome of a chance event
(in a random experiment)
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Flip a Coin 3 times
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Functions of Random Variables
● When you apply a mathematical function g to a random variable X, you create a
new random variable Y = g(X) – why do we need this fact?
● For a random variable it is important to understand its
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probability distribution,
expected value, and
variance
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Probability Distributions (of r.v.)
● A random variable quantifies chance events, and its probability distribution
assigns a likelihood to each of its (r.v.) values.
● Depending on nature of event the r.v. and distribution can be:
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discrete
continuous
● Bernoulli, binomial, exponential, …
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Expected Value and Variance
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Probability Theory
It is about
● modeling real-world phenomena (and understanding what is «normal» and
«unusual»)
● making optimal predictions
● measuring risks/reliability
● drawing inference from «small» samples
● incorporating new evidence
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Probability Space
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Probability Space
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Random Variables and Events
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Probability functions
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Cumulative Distribution Function
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Conditional Probability
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Conditional Probability. Example
Has
Has no
Total
Has
Has no
Total
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Bayes' Theorem
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Bayes' Theorem
● Theorem describes how to update the probabilities of hypotheses when given
evidence
● Philosophically, all probabilities are conditional probabilities
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Part 2. Introduction to Statistics

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Statistical modeling
● Statistical modeling is based on optimization and simulation
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it is important to know optimization techniques
● In this course, we will study various techniques to estimate the parameters, also
by means of resampling and simulation
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The Essence of Statistics
● Statistics solves a backwards problem
● It starts from data (observe) and then asks what was used to generate the data
● With statistics we can quantify predictions
● And also helps to quantify our uncertainty
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Descriptive Statistics
● Descriptive statistics enables us to present the data in a meaningful way, which
allows simpler interpretation of the data
● Typically, two general types of statistic that are used to describe data:
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Measures of central tendency
Measures of spread
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Measuring the Central Tendency
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Measuring the Spread
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Inferential Statistics
● Inferential statistics allows us to use samples to make generalizations about the
populations from which the samples were drawn
● It is, therefore, important that the sample accurately represents the population
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Data and Estimators

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Sample and Population
● In statistics, a population is all of the elements in a group and a sample is a part
of a population chosen to represent the entire population.
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Samples
● A sample should represent main properties of
population (that are investigated in the research /
analysis )
● Reliable statistical analysis deals with representative
samples
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Statistic
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Estimation
● Inferential statistics is focused on the estimation of the population parameter
from the sample statistic
● The sample statistic is calculated from the sample data and the population
parameter is inferred (or estimated) from this sample statistic
● !!! Statistics are calculated, parameters are estimated
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Point Estimates
● The point estimate is the single best guess about the value of parameter
● A good estimator must satisfy three conditions:
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unbiased: the expected value of the estimator must be equal to the value of the parameter
consistent: the value of the estimator approaches the value of the parameter as the sample size
increases
relatively efficient: the estimator has the smallest variance of all estimators which could be used
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Unbiased Estimator. Example
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Maximum Likelihood Estimation (MLE)
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MLE. Example
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MLE. Your task
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Interval Estimates
● A confidence interval contains the true value of the corresponding parameter
with the specified probability
● If we repeat the entire data collection and interval construction process an
infinite number of times (under identical conditions), then 95% of the computed
intervals would contain the true parameter value
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To read before the next class
● Hogg, Robert V., Joseph W. McKean, and Allen T. Craig. «Introduction to
Mathematical Statistics»
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1.1 – 1.9 (to recall the general theory of distribution functions)
2.1 (about conditional probability)
● Hogg, Robert V., Elliot A. Tanis, and Dale L. Zimmerman. «Probability and
statistical inference»
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1.1 – 1.5 (to recall the general theory of distribution functions)
2.1, 2.2 (discrete distributions)
3.1, 3.2 (continuous distributions)
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