"Probability is one of my favorite topics, but I agree with (this part of) the course's treatment."
This statement is an unfortunate juxtaposition: Glad probability is one of your "favorite topics", but you make serious mistakes right at the beginning and show that you don't know anything significant at all about the subject, not at any level from freshman to the texts I listed.
That you "agree" with that "part of the course's treatment" is absurd, especially after what I wrote. That "treatment" is a total upchuck. You, the professor, and CMU should all be humiliated and ashamed. It would be tough to get worse even from a storefront for-profit 'college'. For probability at CMU, I listed an excellent source, Steve Shreve.
"If we really wanted to do probability the right way, we'd tell them that a random variable is a structure-preserving map between measure spaces."
There you go again. You are digging your hole longer, wider, and deeper. You are spouting nonsense.
A "random variable" is definitely not a "structure-preserving map between measure spaces". Not a chance. You won't find "structure-preserving" mentioned anywhere in the definition of a random variable in any of the four texts I listed. Your "structure-preserving" is just gibberish you got from some source you should not touch and then burn outdoors and then flush.
The most advanced definition, as in the four texts I listed, is that a 'random variable' is a measurable function from a probability space into a measurable space. What I described for the set of all events is called a 'sigma algebra'. Then a 'measurable space' is a non-empty set and a sigma algebra of subsets of it. A function is 'measurable' if for each set in the sigma algebra in the range its inverse image under the function is a set in the sigma algebra of the domain. A 'probability space' is a measurable space and a probability measure on that space, and a 'probability measure' is a non-negative measure with total mass 1. For a 'measure', see any of, say,
Paul R. Halmos, 'Measure Theory', D. Van Nostrand Company, Inc., Princeton, NJ.
Walter Rudin, 'Real and Complex Analysis', ISBN 07-054232-5, McGraw-Hill, New York.
H. L. Royden, 'Real Analysis: Second Edition', Macmillan, New York.
I omitted the requirement for a random variable being measurable because for the more important measurable spaces, say, the real numbers with the Borel sets or the Lebesgue measurable sets, finding a function that is not measurable is super tough: The usual construction uses the axiom of choice.
"You can't do that with freshmen."
Well, no one should do "that" with anyone, but with your "agree" with that "part of the course's treatment" here shows that you have been doing even worse, apparently with freshman.
You are flatly refusing to take me at all seriously and refuse to 'get it': Your definition of a random variable is sewage, and you are even defending it.
For what to do with freshman, I gave more than one appropriate, accurate enough, and easy to take, definition of a random variable. There are also other sources. You very much need to take some such source seriously.
You just won't stop digging your hole longer, wider, and deeper: There you go again with your:
"Indeed, we deliberately avoided any mention of continuous probability spaces."
You are spouting more gibberish from some source that should not be touched and then burned and flushed. Your comments continue to fill much needed gaps in the teaching of probability.
There is no such thing as a "continuous probability" space.
Continuity is based on a topology, and there is not necessarily, and usually never is, any topology on the probability space in question. It is true that the Borel sets are the smallest sigma algebra that contain a given topology, usually the 'usual topology' of the real line or Euclidean n-space.
Where continuity enters is in a continuous density or an absolutely continuous cumulative distribution.
In practice what this means is that a 'density' is a continuous function f on, usually, the reals where f is non-negative and its integral is 1. Then the corresponding cumulative distribution F is the 'indefinite' integral of f, in TeX:
F(x) = \int_{-\infty}^x f(x) \; dx
Copy this line into TeX, and the result will look nice, and, more importantly, be correct.
It's crucial to discuss continuous densities in order to discuss random variables with Gaussian, uniform, exponential, chi-squared, etc. distributions. Gaussian is crucial for the central limit theorem. Uniform is crucial for discussing the usual random number generators, e.g., as in the recipes in Knuth's TACP that pass the Fourier test. The exponential distribution is crucial for discussing arrival processes, e.g., when the next Web page request will arrive at a Web server. Chi-square is one of the first distributions encountered in elementary statistical tests, e.g., for independence of two random variables. These distributions are commonly taught in first courses in statistics to students in the social sciences. Looks like the CMU computer science students are falling behind the sociology students!
More details on distributions involve the Lebesgue decomposition and absolute continuity, and you will find solid treatments in each of the four texts I listed along with both Rudin and Royden.
Loève also outlines the bizarre 'singular continuous' case, as I recall, based on the Cantor function.
"Your conclusion may be generally true, but it's definitely off the mark here."
I'm not "off the mark"; I'm dead on target. I gave some good, elementary definitions. Elementary should not mean sewage, but your elementary definition of a random variable is sewage.
"251 is far harder than any undergraduate math course at CMU."
Sounds like the CMU math department doesn't teach, say,
Walter Rudin, 'Principles of Mathematical Analysis, Third Edition', McGraw-Hill, New York.
or the equivalent. Tough to believe.
The material you are teaching is easy, plenty easy enough for a moderately easy course for freshman. Any difficulty is from the need for students to make sense out of sewage content such as your definition of random variable.
Again, for probability done seriously, see the texts I listed; for good elementary texts, there are many, but I have none at hand; often there are good, elementary treatments of random variables in the better texts on statistics for the social sciences; consider also texts on signals in EE; for probability at CMU, see Shreve.
For your course 251, do everyone a big favor and quit teaching it, burn it, and flush the ashes. Then start with some people who actually know the relevant math and develop a good course. And, really, from the list of topics, the course belongs in a math department. And, the course need not be difficult.
Bluntly, computer science very much needs to know math, especially probability, but generally gets a grade of D or less and on probability, an F.
I listed some world class experts in probability (I learned from one of them), but it is true that probability is not popular in US pure math departments, and a significant fraction of math professors will never have seen the more advanced definition of a random variable I have given here; they may not know the strong law of large numbers, conditional expectation, the definition of a Markov process or a martingale, etc.
You are badly wrong. Many people don't know anything about probability; that you don't is not so bad. But that you spout total nonsense about probability is bad; that you defend this nonsense is much worse. Keep fighting me on this and you will dig a seriously big hole for yourself and CS at CMU.
You want me to write Jerry and advise him that 251 needs to be cleaned up?
Yes, and the definition of a measurable map is a function between two measure spaces which takes measurable sets to measurable sets, i.e.: a structure-preserving map between measure spaces. I stopped reading after that. I have no need to defend my mathematical knowledge, and no time to listen to someone who wants me to.
I sincerely apologize for my above post; I did not realize I was dealing with a troll.
"Yes, and the definition of a measurable map is a function between two measure spaces which takes measurable sets to measurable sets, i.e.: a structure-preserving map between measure spaces."
That's not at all what I wrote. You really don't even know how to read a definition in math, do you? Do you know any math at all?
You are wrong again; a counterexample is trivial to construct.
Here you have no need to defend your knowledge of math in general, just on one point, the definition of a random variable.
You are seriously, flatly wrong mathematically. Name calling and refusing to read won't make your nonsense correct.
Enjoy looking like a fool before the world of computing, forever.
This statement is an unfortunate juxtaposition: Glad probability is one of your "favorite topics", but you make serious mistakes right at the beginning and show that you don't know anything significant at all about the subject, not at any level from freshman to the texts I listed.
That you "agree" with that "part of the course's treatment" is absurd, especially after what I wrote. That "treatment" is a total upchuck. You, the professor, and CMU should all be humiliated and ashamed. It would be tough to get worse even from a storefront for-profit 'college'. For probability at CMU, I listed an excellent source, Steve Shreve.
"If we really wanted to do probability the right way, we'd tell them that a random variable is a structure-preserving map between measure spaces."
There you go again. You are digging your hole longer, wider, and deeper. You are spouting nonsense.
A "random variable" is definitely not a "structure-preserving map between measure spaces". Not a chance. You won't find "structure-preserving" mentioned anywhere in the definition of a random variable in any of the four texts I listed. Your "structure-preserving" is just gibberish you got from some source you should not touch and then burn outdoors and then flush.
The most advanced definition, as in the four texts I listed, is that a 'random variable' is a measurable function from a probability space into a measurable space. What I described for the set of all events is called a 'sigma algebra'. Then a 'measurable space' is a non-empty set and a sigma algebra of subsets of it. A function is 'measurable' if for each set in the sigma algebra in the range its inverse image under the function is a set in the sigma algebra of the domain. A 'probability space' is a measurable space and a probability measure on that space, and a 'probability measure' is a non-negative measure with total mass 1. For a 'measure', see any of, say,
Paul R. Halmos, 'Measure Theory', D. Van Nostrand Company, Inc., Princeton, NJ.
Walter Rudin, 'Real and Complex Analysis', ISBN 07-054232-5, McGraw-Hill, New York.
H. L. Royden, 'Real Analysis: Second Edition', Macmillan, New York.
I omitted the requirement for a random variable being measurable because for the more important measurable spaces, say, the real numbers with the Borel sets or the Lebesgue measurable sets, finding a function that is not measurable is super tough: The usual construction uses the axiom of choice.
"You can't do that with freshmen."
Well, no one should do "that" with anyone, but with your "agree" with that "part of the course's treatment" here shows that you have been doing even worse, apparently with freshman.
You are flatly refusing to take me at all seriously and refuse to 'get it': Your definition of a random variable is sewage, and you are even defending it.
For what to do with freshman, I gave more than one appropriate, accurate enough, and easy to take, definition of a random variable. There are also other sources. You very much need to take some such source seriously.
You just won't stop digging your hole longer, wider, and deeper: There you go again with your:
"Indeed, we deliberately avoided any mention of continuous probability spaces."
You are spouting more gibberish from some source that should not be touched and then burned and flushed. Your comments continue to fill much needed gaps in the teaching of probability.
There is no such thing as a "continuous probability" space.
Continuity is based on a topology, and there is not necessarily, and usually never is, any topology on the probability space in question. It is true that the Borel sets are the smallest sigma algebra that contain a given topology, usually the 'usual topology' of the real line or Euclidean n-space.
Where continuity enters is in a continuous density or an absolutely continuous cumulative distribution.
In practice what this means is that a 'density' is a continuous function f on, usually, the reals where f is non-negative and its integral is 1. Then the corresponding cumulative distribution F is the 'indefinite' integral of f, in TeX:
Copy this line into TeX, and the result will look nice, and, more importantly, be correct.It's crucial to discuss continuous densities in order to discuss random variables with Gaussian, uniform, exponential, chi-squared, etc. distributions. Gaussian is crucial for the central limit theorem. Uniform is crucial for discussing the usual random number generators, e.g., as in the recipes in Knuth's TACP that pass the Fourier test. The exponential distribution is crucial for discussing arrival processes, e.g., when the next Web page request will arrive at a Web server. Chi-square is one of the first distributions encountered in elementary statistical tests, e.g., for independence of two random variables. These distributions are commonly taught in first courses in statistics to students in the social sciences. Looks like the CMU computer science students are falling behind the sociology students!
More details on distributions involve the Lebesgue decomposition and absolute continuity, and you will find solid treatments in each of the four texts I listed along with both Rudin and Royden.
Loève also outlines the bizarre 'singular continuous' case, as I recall, based on the Cantor function.
"Your conclusion may be generally true, but it's definitely off the mark here."
I'm not "off the mark"; I'm dead on target. I gave some good, elementary definitions. Elementary should not mean sewage, but your elementary definition of a random variable is sewage.
"251 is far harder than any undergraduate math course at CMU."
Sounds like the CMU math department doesn't teach, say,
Walter Rudin, 'Principles of Mathematical Analysis, Third Edition', McGraw-Hill, New York.
or the equivalent. Tough to believe.
The material you are teaching is easy, plenty easy enough for a moderately easy course for freshman. Any difficulty is from the need for students to make sense out of sewage content such as your definition of random variable.
Again, for probability done seriously, see the texts I listed; for good elementary texts, there are many, but I have none at hand; often there are good, elementary treatments of random variables in the better texts on statistics for the social sciences; consider also texts on signals in EE; for probability at CMU, see Shreve.
For your course 251, do everyone a big favor and quit teaching it, burn it, and flush the ashes. Then start with some people who actually know the relevant math and develop a good course. And, really, from the list of topics, the course belongs in a math department. And, the course need not be difficult.
Bluntly, computer science very much needs to know math, especially probability, but generally gets a grade of D or less and on probability, an F.
I listed some world class experts in probability (I learned from one of them), but it is true that probability is not popular in US pure math departments, and a significant fraction of math professors will never have seen the more advanced definition of a random variable I have given here; they may not know the strong law of large numbers, conditional expectation, the definition of a Markov process or a martingale, etc.
You are badly wrong. Many people don't know anything about probability; that you don't is not so bad. But that you spout total nonsense about probability is bad; that you defend this nonsense is much worse. Keep fighting me on this and you will dig a seriously big hole for yourself and CS at CMU.
You want me to write Jerry and advise him that 251 needs to be cleaned up?