This site uses a number of advanced (but open and standard) technologies, including HTML5, CSS, and JavaScript. To use this project properly, you will need a modern browser that supports these technologies. The latest versions of Chrome, Firefox, Opera, and Safari are the best choices. The Internet Explorer and Edge browsers for Windows do not fully support the technologies used in this project.
Display of mathematical notation is handled by the open source MathJax project."
I've tried the interactive examples also in Safari Technology Preview and Firefox Developer Edition and work ok. :)
Before this becomes one of the many ignored bookmarks in my Favorites folder, can anyone tell me how useful this resource is to a self-ascribed mathematical lightweight as myself?
I took calc BC back in high school and now I'm 31. I realize to a certain extent that many things I believe today are based on a rudimentary understanding of probability theory. Aren't all clinical trials, scientific studies, etc, validated by that 95% confidence interval? And I notice that whenever something happens and my human nature wants draw a conclusion of causality, that cold, rational part of my brain tells me to wait until I have more data.
Anyway I'd love to get into what I feel might be the most fundamental mathematical concept. Probability theory might be our modern day religion.
I looked at it quickly. It has some nice features: Nicely organized with tables of contents, bios of the people mentioned, good references, usually good intuitive introductions to the solid mathematics, some interactive computing examples, good notation, etc.
I especially liked the chapter on sufficient statistics -- the best I've seen, and in my experience not all professors of statistics know this material well. IIRC there is a paper of E. Dynkin that shows that sufficient statistics are not very stable -- I'm not sure yet that the OP covers this.
For your question, it's a good introduction and foundation for work in or that uses probability, statistics, and stochastic processes. Of course, in each case, especially the last two, there is more, e.g., stochastic optimal control. And maybe not all the more recent work in resampling, what Leo Breiman did (used in machine learning, etc.), stochastic differential equations, connections with potential theory, etc. are covered.
But, to answer your question, you sort of need to know what's going on, what the lay of the land is, and that's not so easy to see from the common discussions. I'll try here:
Random Variables: A key, core idea is that of a random variable. So, go out, observe something, get a number, go back. You now have the value of a random variable, call it X. Then X will have a cumulative distribution: For real number x, function F_X(x) = P(X <= x). Here F_X is supposed to be F with a subscript X. So, we use F for cumulative distribution and put the subscript X on it to indicate we're talking about the cumulative distribution of random variable X. The cumulative distribution is simple -- just look at the P(X <= x) part and see that as x increases, that thing grows, cumulatively. So, right, as x increases from -infinity to infinity, F_X grows from 0 to 1, the 1 of certainty.
In the usual cases, X will have an average or expectation, E[X], sometimes -infinity or infinity but usually a finite number. Not all random variables have an expectation -- some goofy, pathological cases don't, but usually don't encounter those in applications.
In nice cases, can take the calculus first derivative (slope) f_X(x) = d/dx F_X(x), and that is the probability density of random variable X. So, the Gaussian bell curve is such a density.
Random variables -- that's the data you work with in probability, statistics, and stochastic processes.
Foundations: For many decades, people had lots of heartburn over the mathematical foundations of probability theory. That was cleared up in 1933 by a paper by A. Kolmogorov, right "father of modern probability theory". Here Kolmogorov used the more fundamental mathematics of measure theory as the mathematical foundations of probability theory. For some decades now, nearly all the more serious work in probability, statistics, and stochastic processes (call those PSSP) has been done using the measure theory foundations. But, often don't need to see the foundations so don't need to confront measure theory.
Measure Theory: You remember calculus, especially the integration part where you find the area under a curve. You did this by partitioning the X axis and getting tall, thin rectangles that under and over approximated the curve. Then you let the width of the widest rectangle go small to zero and took the limit of the areas of the rectangles, the common limit of the over estimate and the under estimate, as the definition of the integral of the curve you started with. Fine. Has worked great quite broadly in pure and applied math, science, engineering, etc. Was invented by Newton but made more precise by B. Riemann and others.
By about 1900, E. Borel and others saw some rough edges with the Riemann integral and cleaned them up. The result is measure theory. Here really measure is just another name for simple old area (leng...
Without looking at your name it only took me about 3 paragraphs to figure out who had written this. I'm glad this particular post drew one of your long responses.
i used this as a resource all through MS while working on a stats second major. it's a very high quality set of notes at the level of Casella Berger (but more complete since it includes measure theory).
Can anyone comment on how this compares to MIT 6.041x Introduction to probability [1]
I started that course a few years ago, but never finished, but really liked it. Wondering if they cover the same, if I should do both, or one instead of the other.
Currently in a MS in Statistics program. This website is definitely on my favorites now. I've been collecting class-contained resources before my start of the program next semester. Here they are, in order of depth/difficulty of the subject:
20 comments
[ 3.3 ms ] story [ 57.9 ms ] threadThis site uses a number of advanced (but open and standard) technologies, including HTML5, CSS, and JavaScript. To use this project properly, you will need a modern browser that supports these technologies. The latest versions of Chrome, Firefox, Opera, and Safari are the best choices. The Internet Explorer and Edge browsers for Windows do not fully support the technologies used in this project.
Display of mathematical notation is handled by the open source MathJax project."
I've tried the interactive examples also in Safari Technology Preview and Firefox Developer Edition and work ok. :)
Hopefully this means I can pass...
(Didn't look too carefully at the content, but it looks good too.)
I took calc BC back in high school and now I'm 31. I realize to a certain extent that many things I believe today are based on a rudimentary understanding of probability theory. Aren't all clinical trials, scientific studies, etc, validated by that 95% confidence interval? And I notice that whenever something happens and my human nature wants draw a conclusion of causality, that cold, rational part of my brain tells me to wait until I have more data.
Anyway I'd love to get into what I feel might be the most fundamental mathematical concept. Probability theory might be our modern day religion.
I especially liked the chapter on sufficient statistics -- the best I've seen, and in my experience not all professors of statistics know this material well. IIRC there is a paper of E. Dynkin that shows that sufficient statistics are not very stable -- I'm not sure yet that the OP covers this.
For your question, it's a good introduction and foundation for work in or that uses probability, statistics, and stochastic processes. Of course, in each case, especially the last two, there is more, e.g., stochastic optimal control. And maybe not all the more recent work in resampling, what Leo Breiman did (used in machine learning, etc.), stochastic differential equations, connections with potential theory, etc. are covered.
But, to answer your question, you sort of need to know what's going on, what the lay of the land is, and that's not so easy to see from the common discussions. I'll try here:
Random Variables: A key, core idea is that of a random variable. So, go out, observe something, get a number, go back. You now have the value of a random variable, call it X. Then X will have a cumulative distribution: For real number x, function F_X(x) = P(X <= x). Here F_X is supposed to be F with a subscript X. So, we use F for cumulative distribution and put the subscript X on it to indicate we're talking about the cumulative distribution of random variable X. The cumulative distribution is simple -- just look at the P(X <= x) part and see that as x increases, that thing grows, cumulatively. So, right, as x increases from -infinity to infinity, F_X grows from 0 to 1, the 1 of certainty.
In the usual cases, X will have an average or expectation, E[X], sometimes -infinity or infinity but usually a finite number. Not all random variables have an expectation -- some goofy, pathological cases don't, but usually don't encounter those in applications.
In nice cases, can take the calculus first derivative (slope) f_X(x) = d/dx F_X(x), and that is the probability density of random variable X. So, the Gaussian bell curve is such a density.
Random variables -- that's the data you work with in probability, statistics, and stochastic processes.
Foundations: For many decades, people had lots of heartburn over the mathematical foundations of probability theory. That was cleared up in 1933 by a paper by A. Kolmogorov, right "father of modern probability theory". Here Kolmogorov used the more fundamental mathematics of measure theory as the mathematical foundations of probability theory. For some decades now, nearly all the more serious work in probability, statistics, and stochastic processes (call those PSSP) has been done using the measure theory foundations. But, often don't need to see the foundations so don't need to confront measure theory.
Measure Theory: You remember calculus, especially the integration part where you find the area under a curve. You did this by partitioning the X axis and getting tall, thin rectangles that under and over approximated the curve. Then you let the width of the widest rectangle go small to zero and took the limit of the areas of the rectangles, the common limit of the over estimate and the under estimate, as the definition of the integral of the curve you started with. Fine. Has worked great quite broadly in pure and applied math, science, engineering, etc. Was invented by Newton but made more precise by B. Riemann and others.
By about 1900, E. Borel and others saw some rough edges with the Riemann integral and cleaned them up. The result is measure theory. Here really measure is just another name for simple old area (leng...
I started that course a few years ago, but never finished, but really liked it. Wondering if they cover the same, if I should do both, or one instead of the other.
[1] https://courses.edx.org/courses/MITx/6.041x/1T2014/info
Stanford https://lagunita.stanford.edu/courses/course-v1:OLI+ProbStat...
CMU http://oli.cmu.edu/courses/free-open/statistics-course-detai...
UCI http://ocw.uci.edu/courses/math_131a_introduction_to_probabi...
http://ocw.uci.edu/courses/math_131b_introduction_to_probabi...
MIT https://ocw.mit.edu/courses/electrical-engineering-and-compu...
Harvard https://projects.iq.harvard.edu/stat110
Textbook: https://www.amazon.com/Stochastic-Processes-Applications-Rob...