Mr. F
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 Mar 19 answered How can I calculate the CDF of this random variable? Mar 19 comment proving gradient of a scalar field is perpendicular to equipotential surface These MIT notes mention the integral definition, but the directional derivative one that you mention is just as good. @Bruno -- fixed the typo, thanks! Mar 19 revised proving gradient of a scalar field is perpendicular to equipotential surface edited body Mar 19 answered Binary random variables event-level independence implies random variable independence Mar 19 revised proving gradient of a scalar field is perpendicular to equipotential surface added 859 characters in body Mar 19 answered proving gradient of a scalar field is perpendicular to equipotential surface Mar 19 revised what is the intuition behind Delta method? added 961 characters in body Mar 19 answered what is the intuition behind Delta method? Mar 19 comment what is the intuition behind Delta method? $f(x) = x(1-x)$, so $f(\mu) = \mu(1-\mu)$. Just follow the exact procedure as in this example, but substituting in the properties of your distribution and your function. Mar 19 comment what is the intuition behind Delta method? You're very close to getting it. First, just plug $\mu$ directly into the formula for $f(x) = x(1-x)$. Think about what the function applied to the mean value is (it's not $k\theta$). Secondly, you are correct to think of the CLT for the approximate distribution of $\bar{X}_{n}$.. but why would it have mean $0$? It won't converge to a standard normal unless you subtract the mean and divide by the standard deviation... so what if you don't do these operations? Mar 19 comment What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6? I've added some Python code that verifies this with eigenvectors and Monte Carlo, and added a bit of discussion. Hopefully it clears things up now. Mar 19 revised What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6? added 3421 characters in body Mar 19 revised What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6? added 3421 characters in body Mar 19 revised What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6? edited body Mar 19 comment What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6? Ahh, also a friend just pointed out that I had an error. The last row should really be $[5/6, 1/6, 0, 0, 0, 0, 0]$, because you don't automatically go back to start if the next play of the game began with a roll of a $1$. That should fix issues with the eigenvector of my transition matrix. I've edited the matrix to reflect that. Mar 19 comment What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6? The idea is that by making the transition matrix entry in the bottom left equal to 1.0, we make the chain as a whole recurrent. In the long run, the chain only sits in the 'accept' state for 1 "time unit" before flipping back to the start. Thus, in the long run, the probability of seeing the chain in that accept state should be equal to the reciprocal of the expected number of rolls to get there. So then you need to use either the system of equations method or the eigenvalue/eigenvector method with the transition matrix to get that long-term probability. Mar 18 revised What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6? edited body Mar 18 revised What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6? added 207 characters in body Mar 18 comment What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6? Ah yes, you're right about that. If you roll a 1 at any stage, you go back to the 1 state. I'll modify the probabilities. Mar 18 revised What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6? edited body