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Probability - The Science of Uncertainty and Data Curso Unit 6: Further topics on random variables Unit overview 1. Motivation
1. Motivation
2. Unit 6 overview
In this unit we discuss a number of topics on random variables:
● Methods for calculating the distribution of a function of one or more random variables, including the special case of the sum of two independent random variables
● The concepts of covariance and correlation between two random variables
● An abstract perspective under which conditional expectations are viewed as random variables
Curso Unit 6: Further topics on random variables Lec. 11: Derived distributions 1. Lecture 11 overview and slides
3. Lecture 11 overview and slides
This lecture develops a method for finding the distribution (PMF or PDF) of a function of one or more random variables with known distribution.
4. The PMF of a function of a discreter r.v.
5. A linear function of a continuous r.v.
6. A linear function of a normal r.v.
7. A PDF of a general function
9. The intuition for the monotonic case
10. The nonmonotonic example
Curso Unit 6: Further topics on random variables Lec. 12: Sums of independent r.v.'s; Covariance and correlation 1. Lecture 12 overview and slides
12. Lecture 12 overview and slides
13. The sum of independent discrete random variables
15. The sum of independent normal r.v.´s
16. Covariance
18. The variance of the sum of the r.v.´s
19. The correlation coefficient
20.Derivation of key properties of the correlation coefficient
21. Interpreting the correlation coefficient
22.Correlations matter
Curso Unit 6: Further topics on random variables Lec. 13: Conditional expectation and variance revisited; Sum of a random number of independent r.v.'s 1. Lecture 13 overview and slides
23.Overview and slides
24.Conditional expectation as a r.v.
26.Stick-breaking revisited
27. Forecast revisions
28.The conditional variance
29.Derivation of the law of total variance
30.A simple example
31. Section means and variances
32.Mean of the sum of a random number of random variables
33. Variance of the sum of a random number of random variables
Curso Unit 6: Further topics on random variables Solved problems 1. The PDF of the absolute value of X
34.The PDF of the absolute value of X
35. Derived distributed example
37. The difference of two independent exponential r.v.´s
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38.The sum of discrete and continuous r.v.´s
39.Using conditional expectation and variance
40.The variance in the stick-breaking problem
41. A coin with random bias
42.Widgets and crates
43.Random number of coin flips
Curso Unit 6: Further topics on random variables Additional theoretical material 1. A linear function of two independent continuous r.v.s
44.A linear function of two independent continuous r.v.s
45. Simulation
46.Conditional expectation properties
Curso Unit 6: Further topics on random variables Problem Set 6 1. The PDF of the logarithm of X
Curso Unit 6: Further topics on random variables Unit summary 1. Unit 6 summary
47. Unit 6 summary
Curso Unit 7: Bayesian inference Unit overview 1. Motivation
48.Motivation
49.Unit 7 overview
In this unit, we focus on Bayesian inference, including both hypothesis testing and estimation problems.
a) We apply the Bayes rule to find the posterior distribution of an unknown random variable given one or multiple observations of related random variables.
b) We discuss the most common methods for coming up with a point estimate of the unknown random variable (Maximum a Posteriori probability estimate, Least Mean Squares estimate, and Linear Least Mean Squares estimate).
c) We consider the question of performance analysis, namely, the calculation of the probability of error in hypothesis testing problems or the calculation of the mean squared error in estimation problems.
d) To illustrate the methodology, we pay special attention to a few canonical problems such as linear normal models and the problem of estimating the unknown bias of a coin.
50.Lecture 14 overview and slides
51. sdfs
