My previous post is about a method to simulate a Brownian motion. A friend of mine emailed me yesterday to tell me that this is useless if we do not know how to simulate a normally distributed variable.

My first remark is: use the rnorm() function if the quality of your simulation is not too important (Later, I'll try to explain you why the R "default random generation" functions are not perfect). However, it may be fun to generate a normal distribution from a simple uniform distribution. So, yes, I lied, I won't create the variable from scratch but from a uniform distribution. 

The method proposed is really easy to implement and this is why I think it is a really good one. Besides, the result is far from being trivial and is really unexpected. This method is called the Box-Muller method. You can find the proof of this method here. The proof is not very complicated, however, you will need a few mathematical knowledges to understand it.

Let u and v be two independent variables uniformly distributed.  Then we can define:

x = sqrt(-2log(u))sin(2 PI v)
y = sqrt(-2log(u))cos(2 PI v)

x and y are two independent and normally distributed variables. The interest of this method is its extreme simplicity in term of programming (We only need 9 lines if we don't want to test the normality of the new variables neither plot the estimation of the density).

We can obtain a vector of variables normally distributed. The Lillie test doesn't reject the null hypothesis of normal distribution. Besides, we can plot the estimation of the density of the variables. We obtain the following plot that looks indeed similar to the Gaussian density.



The program (R):

 # import the library to test the normality of the distribution
library(nortest)

size = 100000

u = runif(size)
v = runif(size)

x=rep(0,size)
y=rep(0,size)

for (i in 1:size){
  x[i] = sqrt(-2*log(u[i]))*cos(2*pi*v[i])
  y[i] = sqrt(-2*log(u[i]))*sin(2*pi*v[i])
}

#a test for normality
lillie.test(c(x,y))

#plot the estimation of the density
plot(density(c(x,y)))


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The financial market is not only made of stock options. Other financial products enable market actors to target specific aims. For example, an oil buyer like a flight company may want to cover the risk of increase in the price of oil.

I found a golden website. The blog of Esteban Moro. He uses R to work on networks. In particular he has done a really nice code to make some great videos of networks. This post is purely a copy of his code. I just changed a few arguments to change colors and to do my own network.

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As you have certainly seen now, I like working on artificial neural networks. I have written a few posts about models with neural networks (Models to generate networks, Want to win to Guess Who and Study of spatial segregation).

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I already talked about networks a few times in this blog. In particular, I had this approach to explain spatial segregation in a city or to solve the Guess Who? problem. However, one of the question is how to generate a good network.

The function apply() is certainly one of the most useful function. I was scared of it during a while and refused to use it. But it makes the code so much faster to write and so efficient that we can't afford not using it.

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Have you ever played the board game "Guess who?".

If you want to choose randomly your next holidays destination, you are likely to process in a way which is certainly biased. Especially if you choose randomly the latitude and the longitude.

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My previous post is about a method to simulate a Brownian motion. A friend of mine emailed me yesterday to tell me that this is useless if we do not know how to simulate a normally distributed variable.

The Brownian motion is certainly the most famous stochastic process (a random variable evolving in the time). It has been the first way to model a stock option price (Louis Bachelier's thesis in 1900).

1

The merge of two insurance companies enables to curb the probability of ruin by sharing the risk and the capital of the two companies.

For example, we can consider two insurance companies, A and B.

How to estimate PI when we only have R and the formula for the surface of a circle (Surface = PI * r * r)?

The estimation of this number has been one of the greatest challenge in the history of mathematics. PI is the ratio between a circle's circumference and diameter.

I was in a party last night and a guy was totally drunk. Not just the guy who had a few drinks and speaks a bit too loud, but the one who is not very likely to remember what he has done during his night, but who is rather very likely to suffer from a huge headache today.

I am currently doing an internship in England. Therefore, I keep alternating between French and English in my different emails and other forms of communication on the Internet. I have been surprised to see that some websites are able to recognize when I use French or when I use English.

The VIX (volatility index) is a financial index which measures the expectation of the volatility of the stock market index S&P 500 (SPX). The higher is the value of the VIX the higher are the expectations of important variations in the S&P 500 during the next month.

The Olympic Games have finished a couple of days ago. Two entire weeks of complete devotion for sport. Unfortunately I hadn’t got any ticket but I didn’t fail to watch many games on TV and internet.

Hello (New World!), 

My name is Edwin, I’m a 22 year-old French student in applied mathematics. In particular, I study probability, statistics and risk theory.

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