Probability and Statistics 1
Weekly outline


https://cesnet.zoom.us/j/97555105547?pwd=VGJSV0RYbGRPeWJvSyt2OUI4NDRQdz09
Meeting ID: 975 5510 5547 Passcode: 501421 
 Lectures in Zoom as long as needed (probably the whole semester). The lecture has its page in Moodle (link in SIS). Everything will be there. (Czech and English classes have different Moodle pages!)
 If there are no technical complications, the video recording of each lecture will be available (after logging in to SIS).
 If you mind being recorded, you can turn off your camera or ask questions in the chat instead of by audio.
 But I'll be happy if you turn on the camera to see how slowly/fast I'm talking, what surprised you, etc.
 Also use the Zoom functions: raise hand, slower/faster.
 We will use short polls during the lecture.
 Pdf version of the ``board'' will also be available  before the lecture, and after with my handwritten comments.
 The exam will ideally be a normal written exam with the possibility of an oral examination. But we will see what the pandemic situation will allow.
 There is also place in Moodle to discuss any issues. Alternatively, contact me by email.


Organization. Definition and examples of prob. spaces. Theorem about basic properties. Conditional probability.


Overview of the last class. Conditional probability: law of total probability (application: Gambler's ruin).
Bayes theorem with applications. Independence. Continuity of probability. (We covered what has handwritten comments on it.)


We studied discrete random variables: PMF's, CDF's, examples of various distributions. We learned two definitions of expectation and the LOTUS rule.


More properties of expectation, variance  linearity, law of total expectation, etc.
Computing expectation of the distributions we have covered.
Teaser for next time: random vectors.


Random vectors (tuples of discrete random variables): joint PMF, independence, expectation of product of indpendent rv's, examples (multinomial distribution, coupling). Proof of linearity of expectation. Convolution formula for the PMF of a sum.
Conditional PMF (start).


Random vectors  conditional PMF. Example: "splitting the Poisson distribution"
Another example  the twoenvelopes paradox. I made a small mistake in the calculation in class, it is corrected in the notes. After correction, it no longer seems to be advantageous to switch the envelopes, which makes the paradox go away. (It seems to return back though when we change the rules of the games, one envelope contains 3x more than the other. But this is a topic for another time ...)
Starting general random variables, particular continuous ones. Definitions of CDF, PDF, some intuition. How to use PDF to find probability of r.v. being in an interval.

I made a small mistake in the calculation in class, it is corrected in the notes.


Computing expectation using PDF, linearity of expectation, variance.
Examples of continuous distributions: uniform, exponential, normal, Cauchy.


Gamma distribution. Universality of uniform distribution.
Continuous vectors: joint cdf, pdf. Highdimensional normal distribution.


Continuous random vectors: conditional distribution, marginals.
Covariance and correlation.
Inequalities: Cauchy, Jensen, Markov, Chebyshev.


Laws of large numbers. Central limit theorem.
Intro to statistics  what type of problems will we look at. What is statistics and what it is not.
Empiric cdf and its properties.


Point estimation: the setup, properties of estimators.
Sample mean and sample variance.
Method of moments, Maximal likelihood.


Another illustration of point estimates.
Interval estimates  based on central limit theorem, based on the Student tdistribution.
Intro to hypothesis testing.


Hypothesis testing.
Goodness of fit (using Pearson chisquare distribution).
Linear regression.


Simpson's paradox.
Permutation test. Bootstrap.
Bayesian statistics.
Briefly about sampling random variables.