Class, Part 2: A. Spanos:
Probability/Statistics Lecture Notes 1: Introduction to Probability and Statistical Inference
Day #1 slides are here.
Class, Part 2: A. Spanos:
Probability/Statistics Lecture Notes 1: Introduction to Probability and Statistical Inference
Day #1 slides are here.
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Very nice, thanks! There is one thing I’m very curious about. Where did Aris get the knowledge from what the “medieval soldiers” knew before Cardano? Are there sources for this?
He may not read this so can you please pass the question on to him?
Thanks!
Christian: Thanks I will. I know it’s in his book.
Catching up on the readings …
The medieval soldiers bit might just be reasonable speculation. Of course, the soldiers would only need to observe that dicers whose lucky number was 7 tended to end up better off than dicers whose “lucky number” was 6 or 8.
The even -versus-odd sum for rolls of a pair of dice doesn’t need the elaborate working out of the 36 possibilities to get a correct answer. If one of the dice has a 50 percent chance of being odd, there is a 50 percent chance that the sum of two dice will be odd. This is true regardless of how many sides the second die has, or what the number pattern is on the second die, or what the probabilities are of rolling each number on the second die.
It is interesting that if the first die is our familiar six-sided die with equal chances of digits 1 through 6, then there is a one-third chance that the sum of two dice will be divisible by 3, regardless of the numbers or chances on the second die. This pattern (1/2 chance of number divisible by 2, 1/3 of number divisible by 3) does not extend to numbers divisible by 4 or 5, but holds for numbers divisible by 6.
This can all be seen by conditioning on the number x shown by the second die, without knowing what x is.
I have no idea whether medieval soldiers, or Cardano for that matter, realized this, although it seems to me more graspable by intuition than the more elaborate working out of the 36 possibilities.
One of the noteworthy issues that arose in our seminar is the difference between the way philosophers tend to talk about probability, namely, in terms of statements corresponding to the occurrence of events, and truth functions “and, or, not..” in relation to them. This contrasts to the corresponding set-theoretic operations and terms, but I cannot remember if I’ve seen a complete “translation”.
In any event, it’s clear that philosophers of probability don’t generally use “random variables”. Terms like (X=x) are considered foreign. But they too could be precisely defined within a formal system of (quantified) logic, with identity. It would be good to demystify random variables. (I’m not saying I’ve succeeded.)
Consider a finite outcomes set S = {s1,s2,s3,…sk}.
A random variable is relative to an event space F: a field associated with the space of events of interest.
A simple random variable X with respect to the event space F, is a function assigning real numbers to each member of S satisfying certain conditions (so as to “preserve the event structure of event space F”.)
A random variable assigns real numbers to the si . Each si gets mapped to a real number by X.
Say S consists of outcomes of two coin tosses:
S = {(T,T), (T,H), (H,T), (H,H)}
If one is interested in the number of “heads” in the 2 coin tosses, random variable X may assign numbers as follows:
X(T,T) = 0
X(T,H) = 1
X(H,T) = 1
X(H,H) = 2
(X = 1) is a shorthand for the set consisting of the members of S that X maps to 1:
{the si that X maps to 1} = {s: X(s) = 1}
So it’s a shorthand for an event.
In general,
(X = x) is a shorthand for {s: X(s) = x}, that is, “the set of si that X maps to x” (lower case x is the value of X)
The above assignment corresponds to 3 events of interest:
A0 ={s: X = 0} = {(T,T)}
A2 = {s: X = 1} = {(T,H), (H,T)}
A3= {s: X= 2} = {(H,H)}
Event space corresponding to X:
F ={ S, { }, {(T,T)}, {(H,H)}, {(T,H), (H,T)}, {(T,T), (H,H)}, {(T,H), (H,T), (H,H)}, {(T,H), (H,T), (T,T)},
Suppose on the other hand that:
X(T,T) = 0
X(T,H) = 1
X(H,T) = 3
X(H,H) = 2
(X = 3) = {(H,T)} but this is not an element of the event space F.
Send thoughts, corrections. For a statistician, I realize, it’s minutia.
For a finite set one would normally use the power set as event space, so that one doesn’t have to bother about sets that may not be member of the event space.
Otherwise it looks alright.
Christian: That was one of the points that arose in trying to define random variables. Once “events of interest” are identified, the corresponding field must reflect it.
Why would one want to restrict the field more than what would be mathematically necessary? (I’m not an expert in quantum physics and rumor has it that there are reasons for doing such things there but I haven’t yet come across a situation in which there were compelling reasons. How can it hurt to be able to handle more probabilities than one is interested in initially?)
Christian: I take the point to be that if one is interested in a subset of events, e.g., # of heads in two tosses, that the values of the random variable must correspond, else the probabilities don’t add up.