The threshold of question 24 represents an average number of how many alerts every day (for a month of 30 days) ?

You write an integer number. A tolerance of 1 unit is taken into account.

Let   be a random variable following a normal distribution of mean   and standard deviation . Using your common sense, what is the VaR at 99.5% of this law ?

For a mixed Nash equilibrium that is not pure equilibrium, what property does it satisfy ?

You consider the random variable which is a loss, and which is described here :

The variable  is negative, which corresponds to a gain, with a probability :

otherwise :

What is the value of the VaR at the threshold 99% of ?

For your answer, you just write an integer number, without any other symbol.

A bank wishes to detect high-value credit card frauds. To do this, she analyzes "extreme value" debits made by customers to determine the threshold at which an alert should be triggered and customer confirmation requested. She receives 177 000 000 transactions every month. In a first examination, you would like to choose a threshold using the "Peak Over Threshold" (POT) method for the amount triggering an alert.

We have seen that a common practice is to keep the  largest values of the data to apply the POT method, and that for this choice of , 3 mathematical formulas are commonly used.

We ask you to choose one of these 3 formulas and propose a number representing the quantity of the most significant transactions to retain for our study.

So, there are 3 possible answers, and all 3 are considered correct for the valuation of this question.

You propose and you write an integer number. We will consider any rounding problems you may encounter.

Be careful, if you find something like 23580, you don't write 23,580 nor 25 580, but 23580.

What is the value of the correlation coefficient of the variables  and  of the preceding question ?

As in question 2, you use a comma if you need it. Two decimal places are enough.

We have  built 1500 blocks of 900 samples by simulating an exponential random variable whose expectancy is 4.8. We have computed the maxima of each 1500 blocks and put them in a vector named

Using a specific library (R or Python), we have fitted the 1500 maxima with a GEV distribution and obtained a distribution function  .

We can see from the results that the value of xi (corresponding to the value  given in the course) is close to a number with few digits after the decimal point. What is the value of this number ? Two decimal places are enough. As in question 2, you use a comma if you need it.

If there is not a pure Nash equilibrium in a game, is it true that there exist a mixed Nash equilibrium ? 

In this question you consider the Expected Shortfall as the  of the course.

If you encounter some difficulties, you can also compute the following quantity : 

What is the Expected Shortfall of at the threshold of 99% ? (both computations and both answers will be considered correct, give just the result).

Here, you write the answer with a comma, e.g. if you find 13.82, you write 13,82. Two decimal places are OK for this question.