How can all events be causally related if there is no initial event?
12114
For a number of reasons &mdash; including perhaps a desire to feel that we have a complete understanding of where we came from, or at least an understanding which is completely sufficient for all of our purposes &mdash; there is a strong tendency to suppose that an infinite regress of causes and effects is impossible. We see this for instance in the . Traditionally in Greco-Roman influenced philosophy, this prime mover has been identified with the notion of a creator god. A more modern formulation comes from the Big Bang theory, which (although we study the question) happened for no reason that we know of; and perhaps occurred for no reason at all. Indeed, if time is a property of the universe, a good argument can be made that .  At the same time, we have gotten a lot of leverage in science from the ideas that everything happens for a reason, i.e.&nbsp;because it is caused by something else. This is sometimes described as the . Aristotle also applies this idea, except that the first mover is made an exception, something which exists literally for no reason. Indeed, the emotionally negative way that "no reason" is used in common speech, as something groundless and irrational, indicates how widespread the feeling is that the idea of uncaused events is deeply dissatisfying.  Almost all of our modes of critical thinking are infused with both of these ideas: that we may work from first principles, a definite starting point, counting up from zero (or one, historically); but also that we may trace the causes of things to some point, and then later ask how that starting point came to be. But the two ideas are themselves in conflict.  Which is true &mdash; that infinite causal chains are impossible? Or that they are necessary? Or are they perhaps possible without being necessary?   Related questions:
90978
I recently learned that two independent random variables $X$ and $Y$ must have a covariance of $0$. That means that the correlation between them is also $0$. However, apparently, the converse is not true. 2 random variables $X$ and $Y$ can have a correlation of $0$, yet still be dependent. I don't understand why this is. Doesn't a correlation of $0$ imply that the random variables do not affect each other?
62710
Let $(\Omega,\mathscr{F},\mu)$ be a probability space, given a random variable $\xi:\Omega \to \mathbb{R}$ and a $\sigma$-algebra $\mathscr{G}\subseteq \mathscr{F}$ we can construct a new random variable $E[\xi|\mathscr{G}]$, which is the conditional expectation.     What exactly is the intuition for thinking about $E[\xi|\mathscr{G}]$?  I understand the intuition for the following:   (i) $E[\xi|A]$ where $A$ is an event (with positive probability).  (ii) $E[\xi|\eta]$ where $\eta$ is a discrete random variable.   But I cannot visualize $E[\xi|\mathscr{G}]$. I understand the mathematics of it, and I understand that it is defined in such a way to generalize the simpler cases that we can visualize. But nonetheless I do not find this way of thinking to be useful. It remains a mysterious object to me.     For example, let $A$ be an event with $\mu(A)&gt;0$. Form the $\sigma$-algebra $\mathscr{G} = \{ \emptyset, A, A^c, \Omega\}$, the one generated by $A$. Then $E[\xi|\mathscr{G}](\omega)$ would be equal to $\frac{1}{\mu(A)} \int_A \xi$ if $\omega \in A$, and equal to $\frac{1}{\mu(A^c)} \int_{A^c} \xi$ if $\omega \not \in A$. In other words, $E[\xi|\mathscr{G}](\omega) = E[\xi|A]$ if $\omega\in A$, and $E[\xi|\mathscr{G}](\omega) = E[\xi|A^c]$ if $\omega \in A^c$.   The part which is confusing is that $\omega \in \Omega$, so why do we not just write $E[\xi|\mathscr{G}](\omega) = E[\xi|\Omega] = E[\xi]$? Why do we replace $E[\xi|\mathscr{G}]$ by $E[\xi| A\text{ or } A^c]$ depending on whether or not $\omega\in A$, but not permitted to replace $E[\xi|\mathscr{G}]$ by $E[\xi]$?     Note. In responding to this question do not explain this by using the rigorous definition of conditional expectation. I understand that. What I want to understand is what the conditional expectation is supposed to be calculating and why we reject one in place of another.
109751
If there is a relation between time and entropy, what is it? Are there limitations for this equation?  Or if there is no relationship between them, what is the current state of research?
57183
In relativity, if two events are simultaneous in a specified frame, they cannot be simultaneous in any other frame.   My question is this: given any two events, is there always a frame in which these two events are simultaneous? For example, if I drop a blue ball on one side of a tennis court, and my friend drops a red ball on the opposite side of the court one day later -- from my frame one day later -- is there a frame in which the blue and red balls hit the ground simultaneously?
101517
Trying to find examples of discrete and continuous random variables that are dependent and uncorrelated.    For the continuous I take standard normal $X \sim N(0,1)$ and $Y=X^2$.  By symmetry it is clear that $E(XY) -E(X)E(Y) = E(X^3) - E(X)E(X^2) = 0 - (0)(1) = 0$, so uncorrelated.   But for events $A = \{0 &lt; X &lt; 1\}$ and $B = \{X^2 &gt; 1\}$, I see that $P(A \cap B) = 0$ which does not equal $P(A)P(B)$ since both $A$ and $B$ have nonzero probability. Thus, $X$ and $Y$ are dependent and uncorrelated.  Now I am having trouble constructing a simple example with discrete random variables.
86125
If we know that two events $A$ and $B$ are independent, can we say that $A$ and $B$ are also conditionally independent given an arbitrary event $C$?  $$P(A\cap B) = P(A)P(B) \overset{?}{\Rightarrow} P(A\cap B|C) = P(A|C)P(B|C)$$
79024
I have a large dataset of tennis-points (who served and if they won etc).  Now I would like to check if the points have any correlation with each other (does winning/losing the previous point affect the current or the one after this?).  I also have an "estimate" of each players serve-winning probability as well as his "actual" serve-winning probability in the game (calculated from the dataset).  Ive tried looking into autocorrelation but I honestly dont know where to start. Does anyone have any idea how to test if Point X have any effect on the outcome of Point X+1?  Any help is greatly appreciated  PS. I tried asking a similar question describing how I thought and the only answer I get was "You are lacking fundamental understanding about probability-theory", hence this time I havent included any of my own thought-process, but feel free to ask for more info if I have excluded something important DS.
123370
The rule says if the order of two actions/events is clear then the past perfect is not necessary, i.e. you can use past simple. So if the sentence has before or after then the order is clear and therefore the rule applies.  says the first sentence is incorrect. Is he/she right? If so, how do you explain what appears, to me, to be contradictory?     If the Past Perfect is not referring to an action at a specific time,   Past Perfect is not optional. Compare the examples below. Here Past   Perfect is referring to a lack of experience rather than an action at   a specific time. For this reason, Simple Past cannot be used.      Examples:      She never saw a bear before she moved to Alaska. Not Correct      She had never seen a bear before she moved to Alaska. Correct   They say "a lack of experience rather than an action". Saying that just does not solve the problem I have which is the past simple fits the rule of order of events. Not seeing is a negative event with the verb "to see".  I hope the answer won't be this is how we speak or this is how English works or since when languages are logical!
29404
Two events are mutually exclusive if they can't both happen.  Independent events are events where knowledge of the probability of one doesn't change the probability of the other.  Are these definitions correct? If possible, please give more than one example and counterexample.
55788
Causality is generally accepted to be a fundamental physical principle. But quantum mechanics is acausal (e.g. there is no 'why' as to the result of a measurement of the position of a particle in an infinite 1D square potential well). In which case what is a distinction between causality and determinism, that has a meaningful non-acausal definition of causality.  It seems to me that "causality = determinism + quantum mechanics". It seems that it doesn't provide any new concepts, it's just a flawed attempt to reconcile determinism with the arbitrary probabilistic nature of QM, which are ultimately unreconcilable. Does anyone agree with this idea? If not, I would like someone to try and convince me that I'm wrong.  I am fully aware that a lot of threads barking up this tree, and I have tried reading many of them, but none seem to crystallize my problem very well.
46380
Seems this would not be a deterministic thing, or is there a way to do this reliably?
26648
$A$ unless $B$ surely means, 'given that $B$ does not happen, $A$ will happen'.  So if $B$ happens, $A$ does not happen.  Yet I've read, by those officially accredited, that $A$ unless $B$ = $A$ or $B$. However, if $B$ happens, and $A$ still happens despite the fact, that this is apparently true.  Surely, the sheffer stroke is appropriate here? So that: $A$|$B$.  If $B$ occurs, $A$ does not, thus, $A$ unless $B$. If $B$ does not occur, $A$ does.
65339
I'm confused about how to reconcile the probability of independent events not having anything to do with prior history, but sequences of events do (seemingly) take into account prior history. This question asks a similar question: . However, having read that, I found I had a very specific confusion about the seeming contradiction between two formulas for probabilities that seem equal to me, but will produce different results based on our understanding of P of sequences versus P of independent events: (A) P(HHHHH) = 0.03125 (B) P(H | HHHH) = 0.5 Can anyone explain how the left side of both equations, P(HHHHH) and P(H | HHHH) are different. And does anything change if we shift from a frequentist to bayesian perspective?
106982
I pursue only intuition; please do not answer with  or . I comprehend the following explanation, but it still does not supply the intuition and I sense a deeper, directer, more intuitive explanation. So how else can I intuit this? Or have I neglected something below? I rewrote: p 51-52.  (2010 2 ed) by Henle, Garfield, Tymoczko.    If the kid misbehaves and the teacher forces him to stay, then the teacher spoke truthfully. Suppose the kid misbehaves but travels. Then clearly the teacher lied. But what if the kid misbehaves, but a big snowstorm forces the trip’s cancellation? You can’t say the teacher lied. The kid behaved but didn’t go; nobody went. The teacher spoke truthfully. 1. If the kid behaved and travels, the teacher spoke truthfully.  Notice that the pattern of Ts and Fs we get, is the pattern for disjunction (see p. 38).
56007
I'm trying to find an example for $n$ events such that each $k$ ($k &lt; n$) are independent, but the $n$ events together are dependent.  Two events are independent if the probability for both of them to happen equals to the product of the probability of each one separately.
103219
Is there a term for the coincidence that occurs when an issue ONLY occurs until the moment that you ask someone else to watch or help you resolve it?    For example, having an issue with software until you try to show someone, then it stops being problematic.  Or an issue with the computer/phone that resolves as soon as a help ticket is put in.    The final example is when you have a problem with your car (funny noise, etc) - you hesitate taking it in for weeks because you know it will disappear as soon as the mechanic drives it...then it does.    This is different than a , where the action of applying a debugger, for example, changes the behavior of the activity being watched.  It's also different from a 'works on my machine' scenario, where only the original user has the problem.   What is it called when a problem resolves itself just by the mention of it?  Jinxing the issue? :)  I see this every day supporting my software product -- the users are experienced and are not making mistakes at one time then doing something different when I watch.  The problems also truly resolve (coincidentally) as soon as they ask for help - they do not reoccur.  I guess it's a case of synchronicity?