On Divisions - student

The Baron's game most recent game consisted of a series of some six wagers upon the toss of an unfair coin that turned up one side nine times out of twenty and the other eleven times out of twenty at a cost of one fifth part of a coin. Sir R----- was to wager three coins from his purse upon the outcome of each toss, freely divided between heads and tails, and was to return to it twice the value he wagered correctly.

Clearly, our first task in reckoning the fairness of this game is to figure Sir R-----'s optimal strategy for placing his coins. To do this we shall need to know his expected winnings in any given round for any given placement of his coins.

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Annealing Down - a.k.

A few years ago we saw how we could search for a local minimum of a function, being a point for which it returns a lesser value than any in its immediate vicinity, by taking random steps and rejecting those that lead uphill; an algorithm that we dubbed the blindfolded hill climber. Whilst we have since seen that we could progress towards a minimum much more rapidly by choosing the directions in which to step deterministically, there is a way that we can use random steps to yield better results.

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Further On A Calculus Of Differences - student

As I have previously reported, my fellow students and I have found our curiosity drawn to the calculus of sequences, in which we define analogues of the derivatives and integrals of functions for a sequence sn with the operators

  Δ sn = sn - sn-1

and
  n
  Δ-1 sn = Σ si
  i = 1
respectively, where Σ is the summation sign, for which we interpret all non-positively indexed elements as zero.

I have already spoken of the many and several fascinating similarities that we have found between the derivatives of sequences and those of functions and shall now describe those of their integrals, upon which we have spent quite some mental effort these last few months.

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Copulating Normally - a.k.

Last year we took a look at multivariate uniformly distributed random variables, which generalise uniform random variables to multiple dimensions with random vectors whose elements are independently uniformly distributed. We have now seen how we can similarly generalise normally distributed random variables with the added property that the normally distributed elements of their vectors may be dependent upon each other; specifically that they may be correlated.
As it turns out, we can generalise this dependence to arbitrary sets of random variables with a fairly simple observation.

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Divisions - baron m.

Greetings Sir R-----! I trust that I find you in good spirit? Will you join me in a draught of this rather fine Cognac and perchance some sporting diversion?

Good man!

I propose a game that ever puts me in mind of an adventure of mine in the town of Bağçasaray, where I was posted after General Lacy had driven Khan Fetih Giray out from therein. I had received word that the Khan was anxious to retake the town and been given orders to hold it at all costs.

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The Cumulative Distribution Unction - a.k.

We have previously seen how we can generalise normally distributed random variables to multiple dimensions by defining vectors with elements that are linear functions of independent standard normally distributed random variables, having means of zero and standard deviations of one, with

  Z' = L × Z + μ

where L is a constant matrix, Z is a vector whose elements are the independent standard normally distributed random variables and μ is a constant vector.
So far we have derived and implemented the probability density function and the characteristic function of the multivariate normal distribution that governs such random vectors but have yet to do the same for its cumulative distribution function since it's a rather more difficult task and thus requires a dedicated treatment, which we shall have in this post.

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On Turnabout Is Fair Play - student

Last time they met, the Baron challenged Sir R----- to turn a square of twenty five coins, all but one of which the Baron had placed heads up, to tails by flipping vertically or horizontally adjacent pairs of heads.
As I explained to the Baron, although I'm not at all sure that he was following me, this is essentially the mutilated chess board puzzle and can be solved by exactly the same argument. Specifically, we need simply imagine that the game were played upon a five by five checker board...

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Multiple Multiply Normal Functions - a.k.

Last time we took a look at how we could define multivariate normally distributed random variables with linear functions of multiple independent standard univariate normal random variables.
Specifically, given a Z whose elements are independent standard univariate normal random variables, a constant vector μ and a constant matrix L

  Z' = L × Z + μ

has linearly dependent normally distributed elements, a mean vector of μ and a covariance matrix of

  Σ' = L × LT

where LT is the transpose of L in which the rows and columns are switched.
We got as far as deducing the characteristic function and the probability density function of the multivariate normal distribution, leaving its cumulative distribution function and its complement aside until we'd implemented both them and the random variable itself, which we shall do in this post.

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On A Calculus Of Differences - student

The interest of my fellow students and I has been somewhat piqued of late by a curious similarity of the relationship between sequences and series to that between the derivatives and integrals of functions. Specifically, for a function f taking a non-negative argument x, we have
  x
  F(x) = f(x) dx
  0
  f(x) = d F(x)
  dx

and for a sequence s having terms

  s1, s2, s3, ...

we can define a series S with terms
  n
  Sn = s1 + s2 + s3 + ... + sn = Σ si
  i = 1
where Σ is the summation sign, from which we can recover the terms of the sequence with

  sn = Sn - Sn-1

using the convention that S0 equals zero.
This similarity rather set us to wondering whether we could employ the language of calculus to reason about sequences and series.

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Every Which Way Is Normal - a.k.

A few months ago we saw how we could generalise the concept of a random variable to multiple dimensions by generating random vectors rather than numbers. Specifically we took a look at the multivariate uniform distribution which governs random vectors whose elements are independently uniformly distributed.
Whilst it demonstrated that we can find multivariate versions of distribution functions such as the probability density function, the cumulative distribution function and the characteristic function, the uniform distribution is fairly trivial and so, for a more interesting example, this time we shall look at generalising the normal distribution to multiple dimensions.

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