Selection_189

Somewhere else, part 151

Some writing worth reading, found here and there

see also

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Selection_166

Somewhere else, part 150

Some writings worth reading, here and there

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Selection_117

London, Bayes and the Lloyd’s

Monday, we really had a great conference in London.

It was a great pleasure since I did learn a lot of things. And also a great honor to be the last speaker. Tuesday morning, I wanted to go to Thomas Bayes’grave, which is the the graveyard next to the CASS Business School. I had a good a apriori about where the grave should be,

but to be honest, it was not possible to get close enough to be able to read the name on it (even if I now know that it is the large one in the right lower corner of the picture)

Actually, on the internet, you can find some picture where the stone is clean, so you can learn that the grave is the “cotton” one – at least, you can easily read that name.

It was actualy more simple to see William Blake’s grave, as well as Daniel Defoe’s.

Then, with Leo, we’ve been to the Lloyd’s to see some friends, as well as Richard Rogers’s building.

At the 11th floor, you have a lot of rooms for meetings, as well as old paintings, to tell a bit more about the history of the company,

The building is just amazing. Unfortunately, to get in, there is a dress code. A sort of strict one actually. Leo is working for RBC, so he casually wears a suit. But I don’t. I mean, I did have a shirt, but as someone mentioned, “there is no collar !” (I don’t want to put my friend into trouble for helping me getting in).

So, after going throught the basement, we’ve been able to reach the elevator, and go on top.

The building is not exactly located where Edward Lloyd got his coffee shop (even after moving at the end of 1691 on Lombard street), but the Lloyd’s is still a legend for anyone interested in the history of insurance, and more generally, the history of risk modeling (and management).

Somewhere else, part 148

Some writings worth reading

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Somewhere else, part 147

Some writings worth reading

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Selection_125

Variance of the Average of a Sequence

In the case where  are i.i.d. random variables, then

Now, what if  are identically distributed, but no longer independent. What if we have an autoregressive process? Assume that

Then

can be written

Here, we will express the variance as a function of  and , but it is possible to use also , since, in the context of an ,

Now, since  we get

which can be simplified, since

i.e.

So, the variance of the mean can be writen as

Observe that if  is large enough,

This asymptotic relationship is well known actually. A simple way to get it is the following. One can can write

or equivalently

But actually, the first relationship is probably more interesting to get an asymptotic approximation,

In the context of an  process, this can be writen

Thus, we get the following well-known relationship

In the case where  is an i.i.d. sequence, i.e. , then we get the relationship mentioned initially. And in the case of a random walk… unfortunately, we cannot use that relationship. But observe that

i.e.

which can be written

If we compare the true value and the approximation, we get the following graph,

> V=function(phi,s2=1,n=100){
+ g0=s2/(1-phi^2)
+ if(phi<1){
+ if(phi==0){v1=g0/n}
+ if(phi>0){v1=g0/n^2*(n+2*((n-1)*
+ phi^(-1)-n+phi^(n-1))/(phi^(-1)-1)^2)}
+ v2=g0/n*(1+phi)/(1-phi)
+ }
+ if(phi==1){
+ v1=(2*n+1)*(n+1)*s2/(6*n)
+ v2=NA
+ }
+ return(c(v1,v2))}
> 
> Vphi=function(phi) V(phi,1,100)
> x=seq(.01,1,by=.02)
> M=matrix(unlist(lapply(x,V)),nrow=2)
> plot(x,M[1,],type="l",col="red",log="y",
+ ylab="Variance of the average (log scale)",
+ xlab="Autoregressive coefficient")
> lines(x,M[2,],col="blue")

Somewhere else, part 146

Nothing special about 146. But some writings worth reading,

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Somewhere else, part 145

Not only is 145 a Leyland number, since 145=34+43, much more fun, it is a factorion, i.e. it is the sum of the factorials of its digits 145=1!+4!+5! which is quite a rare property actually (the other known one being 40,585… so the next time I will mention this property on my blog is probably in 400 years).

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Somewhere else, part 144

144=24· 32 is the twelfth Fibonacci number (and the largest one to also be a square), and the maximum determinant in a 9×9 matrix of zeroes and ones is precisely 144 (source http://wikipedia…)

and on top of that nice cartoon by @RinaPiccolo, some writings worth reading

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Somewhere else, part 143

(via http://akeppleaday.tumblr.com/88690322237 …). Today, some writings worth reading, discovered somewhere else,

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