What is the distribution of the following event?

  • Thread starter Thread starter number0
  • Start date Start date
  • Tags Tags
    Distribution
number0
Messages
102
Reaction score
0

Homework Statement



See uploaded imagery.


Homework Equations





The Attempt at a Solution



See uploaded imagery.


I know that the distribution has something to do with the 1-dimensional Ising model being sampled via Gibbs sampler method, but that is all I know. Anyone have a hint/solution? Thanks.
 

Attachments

  • hw.jpg
    hw.jpg
    25.5 KB · Views: 412
Physics news on Phys.org
For any a, b, c \in \{-1,1\} we have
P(x=a|y=b,z=c) = \frac{P(x=a \cap \{y=b,z=c\})}{P(y=b,z=c)} = \frac{P(x=a,y=b,z=c)}{P(y=b,z=c)},
where P(y=b,z=c) = \sum_{w=-1}^1 P(x=w,y=b,z=c). This has nothing to do with Physics or the Ising model or anything else like that; it is just elementary probability.

RGV
 
Sorry, but I already got your answer (see imagery)... it's just that I do not know what distribution the conditional distribution follows...
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top