Accredited Ph.D. course

**Lectures delivered by
Ernő Keszei, emeritus professor of chemistry**
,
Room No. 148, phone: 372-2500 / ext. 1904, __
keszei-AT-chem.elte.hu__

Time and location:
Tuesday 4:00 to 5:30 PM, Room 132, Chemistry Building

**Exam dates:**
to be discussed; in December and January

**Exams are forseen in** **
Room No. 148, Chemistry Building**

Actual schedule:

September |
October |
November |
December |
|||

3 | 7 | 5 | ||||

10 | 14 | 12 | ||||

17 | 21 | 19 | ||||

19 | 24 | 28 | ||||

26 | 31 |

**September **
12,
**
Tuesday:**

Registration week

**September **
19,
**
Tuesday:**

Negotiation on lecture time

**September ****
26,
Tuesday:**

**1 ^{st}
lecture:**
Probability theory basics I. - Random experiment, outcome, sample space, event.
Postulates (axioms) of probability theory and some properties following from
them. Independence and conditional probability. Probability of independent
simultaneous events. Interpretations of prbability: frequency, classical
probability and geometric probability. Random variables. Simple events concerning
discrete and continuous random variables. Sampling distributions. Properties of
the probability density function. Relation of the probability distribution
function to the probability density function. Calculation of probabilities based
on the probability distribution function and on the probability density
function.

To read:

Suggested literature: Paradoxes in Probability Theory and Mathematical Statistics. Auxiliary material: Probabilty basics demo.

**October****
3,
Tuesday:**

**
2 ^{nd} lecture: ** Probability theory basics II. -
Expectations and their properties. Calculation of the expectation as a
linear operation. Some particularly important expectations. (Distribution mean,
distribution moments, variance, covariance, correlation coefficient, entropy.)
Some important relations for calculating expectations.
The covariance matrix and its properties.
Relation between independence of random variables and their covariance.
Calculation of (normalised) probability density function, if not known. The law
of large numbers. Stochastic convergence and random walk.

**
October
10,
Tuesday: **
**
3 ^{rd} lecture:** Probability theory basics III. -
Some important
probability distributions. The binomial and hypergeometric distributions.
Poisson process: interrelation of the uniform, the Poisson and the exponential
distributions.
Gamma distribution. Normal distribution.

Homework:

**October ****17,
Tuesday:**

**4 ^{th} lecture:**
Probability theory basics IV. -
Properties and use of the Normal distribution.
Construction of a p.d.f. from physical calculations.

Homework:

**October ****24,
Tuesday:**

**5 ^{th} lecture: **
Introduction to statistical methods. Chi-squared, Student's t, and Fisher's F
distribution. Distributions without a maximum. The arcsin distribution.

The aim and methods of statistics. Population vs. sample. Sampling. Estimation and characteristics of estimators. Estimation methods. Histograms. Sample statistics. Sample mean, sample variance and covariance.

Auxiliary material: Problems concerning sampling in a Hungarian election and a U.S. election. A short appetizing paper and a deeper analysis on cognitive bias (or "self fooling") from Nature.

Homework:

**October ****31,
Tuesday:
**

**
November 7,
Tuesday:
**

Auxiliary material: Assignment for histogram construction.

**
November 14, Tuesday:
7 ^{th} lecture: **
Estimation: The method of least squares. The method of moments. Other estimation
methods. Estimation of expectation and variance of functions of random
variables.

Confidence intervals. Formulation of confidence in a computable form. Confidence interval for the expectation of a normal distribution with known variance. Confidence interval for the expectation of a normal distribution with unknown variance. Confidence interval for the the parameters of a binomial distribution.

Auxiliary material: Assignment for parameter estimation using the method of moments.

**
November
21, Tuesday:
**
Confidence interval for the the parameters of a binomial distribution.
Confidence interval for the variance. Approximate confidence interval for a
function of random variables. Confidence interval for the difference of the
expectations of two random variables.

8^{th} lecture:

Statistical hypothesis testing - general considerations. Null hypothesis and alternative hypothesis. Types of hypotheses. Statistics underlying decision making. Type I and type II errors. Power function of the test.

Auxiliary material:

**
November 28, Tuesday:
9 ^{th} lecture: ** Statistical hypothesis testing:
Null hypothesis and alternative hypothesis. Types of hypotheses. Statistics
underlying decision making. Type I and type II errors. Test on the mean of a
normal distribution with known and unknown variance. Test on the parameter

Auxiliary material:

**
December
5, Tuesday:
**
Tests on matching pairs. Nonparametric tests. The Sign-test; the Mann-Whitney
and Wilcoxon (Rank-Sum) test. Tests on several means: One-way ANOVA and
two-way ANOVA tests. Multivariate analysis of variance (MANOVA).
Homoscedasticity tests (tests on variances). Functional relations between random
variables. Testing the correlation coefficient.

10^{th} lecture:

Auxiliary material:

**
December 12,
Tuesday:
** Estimation of parameters
describing functions of random variables. The general straight line and the
straight line through the origin. Testing the difference between the two cases:
the significance test of the intercept. Weighted least squares (LSQ) estimation.
Optimisation of weights to give MVU estimation. Implicit regression.
Overview of the conditions for the validity of LSQ estimation.
Multivariate analysis: a short overview of multivariate methods.

11^{th} lecture:

**
December 19,
Tuesday:
**

No lecture

**
**