## Mathematics

### Exercise 0.5 (Simple probabilities)

There is a round table with 5 seats, and 5 people born on different days. How many ways are there to seat them so that their ages are ordered?

### Exercise 1 (Basic probabilities)
Mike has 2 kids. If we know that at least one of them is a boy, what is the probability that the two of them are boys?

### Exercice 1.5 (Basic probabilities)
An opaque bag contains 2 black balls and 2 white balls. You randomly draw one without looking at it and put it aside. You then draw a second ball and see that it is black. What is the probability that the first ball is white?

### Exercise 2 (Math and computing knowledge)
Give an approximate value of sqrt(2570000).

### Exercise 3 (Math related to data science)
Explain what a hyperplane is (in 2 sentences maximum).

### Exercise 4 (Mathematical reasoning)
The plane contains a finite number of dots of two different colors (two dots with a different color can be in the same location). Between two points of the same color lies a point of the other color. What can you say about the location of the points?

### Exercise 5 (Probabilities)
You have a choice between two games. In both games, a coin is tossed until a certain pair of sides appears successively: heads followed by heads (game 1), or heads followed by tails (game 2). When the pair appears the game stops and you gain 1/n €, where n is the number of coin tosses. For example, if you are playing game 1, the sequence of tosses can be: THTHH and you win 1/5 €. Is there a game which makes you richer on average?

### Exercise 6 (Probabilities)
Consider a country where 40% of the inhabitants live in the northern part, with the rest living in the southern part. In the summer, 30% of the northerners leave the country for vacation, but only 15% of the southerners do so. If, outside of this country, you meet an inhabitant from this country, what is the probability that they are a southerner?

### Exercise 7 (Statistics)
Let’s say we have an estimate of a conversion rate (number of purchases / number of calls) with a statistical accuracy of 1 %, with 15 000 calls. How many calls do we need in order to have a statistical accuracy ten times better (0.1 %)?

### Exercise 8
Through A/B testing, we get the following results:

|               | Test group           | Control group  |
| ------------- |:-------------:| -----:|
| Group size     | 117 415 | 117 284 |
| Conversion rate observed     | 7.07%     |   9.36% |

What is the confidence interval of the incremental gain (Test conversion rate - control conversion rate)?

### Exercise 9 (More difficult problem)
Arthur chooses a secret polynomial with non-negative integer coefficients. You can give him a real number, and he will give you the value of the polynomial at this number. Find a way of determining the polynomial by asking Arthur for such a value a minimal number of times. [Hint: a polynomial in x with non-negative integer coefficients can represent a number expressed in base x, for x large enough.]

### Exercise 10 (More difficult problem)
Put N dots in a circle. Select a starting dot (let’s call it #1). While always going in the same direction from this starting dot:

- remove the next remaining dot (at the beginning, this is the dot following the starting dot: dot #2),
- move to the next remaining dot (at the beginning, this is dot #3),
- iterate (remove, move).

At the end, only one point remains: which one? At first, find a way of calculating this quickly enough (and calculate, for instance, the remaining dot, starting with 20 dots). [Hint: recurrence.] Then, find a way to calculate the last remaining dot in one step, that uses binary.


### Exercise 11 (Calculus)
Give an approximate value of sqrt(10) with 3 significant digits. Give an approximate value of ln 3 with 2 significant digits. [Hint: Taylor expansion]



## Computer Science

### Exercise 1 (C, C++,…)
What is passing arguments by reference?

### Exercise 2 (Python)
What concepts are important for argument passing in Python? Or: Python does not have passing by reference, but it does have something similar, what is it?

### Exercise 3 (Databases)
In relational databases, what is a primary key?

### Exercise 4 (Time complexity)
Given an index, what is the time complexity of retrieving an element in an array? In a Python list?

What about deleting this element?

Given a key, what is the time complexity of retrieving the associated value from a hash table (which implements an associative array, as is the case for instance in Python)?

What about deleting this key?

### Exercise 5 (Time complexity)
What is the complexity of a matrix multiplication?

### Exercise 6 (Shells: Unix,…)
In a Unix shell, what does the shell pipe "|" do?
What does `LD_LIBRARY_PATH=` in the following command do (this is not a question about `LD_LIBRARY_PATH` but a question about the syntax used, what it does, etc.)?

`LD_LIBRARY_PATH= awk… `

### Exercise 7 (More advanced programming)
In object-oriented languages, what is polymorphism?


## Python

### Exercise 1
Do objects have a type, in Python?

### Exercise 2
What is a list comprehension? What is a dictionary comprehension?

### Exercise 3
In a Python shell, what does the variable called “_” (underline) contain?

### Exercise 4 (data science)
Cite a few plotting libraries in Python. Cite a dataframe manipulation library in Python. Cite a machine learning library in Python.

### Exercise 5 (special character handling)
Explain what are the encoding and decoding of characters. What do you know about Unicode? UTF-8? UTF-16?

### Exercise 6 (more advanced)
What does the heapq module do? In which situation is it useful?

### Exercise 7 (more advanced)
What is a *staticmethod*? What is a *classmethod*?

### Exercise 8 (for pros)
In what circumstances is the variable called “_” (underline) typically used in a Python program?

## Machine learning

### Exercice 0 (Basic)

Do you normalize variables before doing machine learning?

### Exercise 1 (Model fitting)
Explain what overfitting is in machine learning (in 2 sentences maximum). How do you see that a model is overfitting?

### Exercise 2 (Model fitting)
What is regularization and what does it do, when fitting a model?

### Exercise 3 (Basic probabilities and fitting)
What is the principle behind linear regression? Prove the formula that gives the parameters of the regression.

### Exercise 4 (Non-linear Support Vector Machines)
In Support Vector Machines, what is the kernel trick?

### Exercise 5 (More advanced probabilities/modeling)
What is the log-loss function: formula, including in the case of more than two classes? In what cases should it be used (instead of, say, a sum of squares error)?

## Business and Insurance

### Exercise 0

What are typically the different steps of the work of a data scientist on a business project?

### Exercise 1
What is an uplift model?

### Exercise 2
How would you test a targeting model in real life?

### Exercise 3 (Big Data/Insurance)
How do you think how Big Data can have an impact in the insurance sector?

### Exercise 4 (Insurance)
What are the 3 biggest business lines in the insurance sector?

### Exercise 5 (Insurance)
What happens if an insurance company gives the same contract price to everybody?

### Exercise 6 (Insurance)
How would you price a contract?

### Exercise 7
I have tested the conversion rate of my cross-sell model using an AB test (randomized test). I get the following results:

|               |	Test group	  |Control group|
| ------------- |:-------------:| -----:      |
|Group size	| 117 415          | 	117 284    |
|Conversion rate observed	     |7.07%|9.36%|

What is the confidence interval of the incremental gain (TEST conversion rate - CONTROL conversion rate)?