Recent content by SchroedingersLion

Hi people, not sure whether I am the only one but when using the "preview" feature after writing up a thread, it does not show the compiled Latex equations. This means I have to post the thread first and correct the various Latex errors via multiple edits (which is not a problem for me, but it...

Greetings! I believe I found an error in a paper to Bayesian neural networks. I think the expression of the covariance of the posterior predictive is wrong, and I wrote down my own calculation. Would be great if a seasoned Bayesian could take a look. Imagine a regression scenario. We want to...

Thank you. Writing the integration domain explicitly in terms of the variable helps to clear my confusion. I need to replace everything ##z##-related with the corresponding ##x## expression, not just a part of the whole thing.

Greetings all. I just got confused by the following. Consider volume integral, for simplicity in 1D. $$ V(A) = \int_{A} dz. $$ If ##z## can be written as an invertible function of ##x##, i.e. ##z=f(x)##, we know the change of variables formula $$ V(A)=\int_{A} dz= \int_{z^{-1}(A)} |z'(x)|dx...

Thanks Ibix!

OK, one can assume that the plane's engine speed ##v## is always measured relatively to the medium. In that case, with tailwind, one has ##t_1=\frac {s}{v+w}## and with headwind ##t_1=\frac{s}{v-w} ##. Adding them, one sees that the resulting time is larger than the time without wind, i.e...

Hi everyone. I came across the following brainteaser: A plane travels from airport A to airport B and then returns to A from B. There is no wind, both trips follow a straight line, and the plane flies at constant engine speed. Suppose now that a constant wind is blowing from A to B. Will the...

Thanks for the response. I just don't understand how the entropy of the constrained system is connected to the unconstrained system. I have ##N## total particles and a total volume ##V## that is split between two subvolumina, ##V=V_1+V_2##, where ##N_1## particles are in ##V_1##, and ##N_2##...

Hi everyone, I have a fundamental question to the first part of Swendsen's Intro to StatMech and Thermodynamics (book). Suppose we have two isolated systems of volumes ##V_1## and ##V_2##. We distribute ##N## ideal gas particles across the two systems with total energy ##E##. Suppose we bring...

Thanks Chestermiller! The author did not ask for the displacement, so it is really unclear what he even expects. Glad to know that I did not miss something obvious.

Greetings! Can you help me understand what this text book problem asks of me? The situation was considered in the text: In equilibrium, the forces on both sides of the piston are equal: ##A_1P_1 = A_2P_2##. This is the first equation. It should also answer part 3 of the question. The piston...

Right... I understand that both the quations ##P(E|E_A)=P(E_B)## and ##P(E)=P(E_B|E_A)## make sense under the assumption that the constraint ##E=E_A+E_B## holds. I am just not used to being able to "choose" between two independencies. But it makes sense that if I have three systems and only a...

So if I know a priori that there are formally three systems ##(E,N), (E_A, N_A), (E_B,N_B)## with ##E=E_A+E_B## and ##N=N_A+N_B##, and now I am given ##E_A##... I would say that ##P(E|E_A)=P(E_B|E_A)##. If I now assume that ##E_B## and ##E_A## are independent, then yes, we have...

Bayes rule for two events ##A,B##: $$ P(A|B) = \frac{P(B|A)P(A)}{P(B)} $$ So written for my events in my previous post: $$ P(E_A | E) = \frac{P(E | E_A)P(E_A)}{P(E)} $$ In order to obtain the desired shape given by @vanhees71 we need to have that ##P(E | E_A)=P(E_B)##. Does this make sense?

Thank you Vanhees, I think that's what I need. Even though I still don't understand why your second formula is obvious. Rewritten: $$ P_{part}(E_A)P(E) = P(E_A)P(E_B) $$ ##E_A## is the event of having ##N_A## particles at energy ##E_A##. ##E_B## is the event of having ##N-N_A## particles at...