First some general information regarding the exam. For problems with multiple parts the points were assigned as follows:

Problem 3 - a: 3p, b: 7p.

Problem 4 - a: 7p, b: 3p.

Problem 5 - a: 1p, b: 1p, c: 4p, d: 4p.

The second part of problem 4 was a bit vague on the exam. Although the problem formulation was clarified during the exam, this has been taken into account in the grading process.

Grades have been assigned according to the following scores (incl. bonus points):

A: > 39

B: 35 - 39

C: 31 - 34

D: 27 - 30

E: 24 - 26

Fx: 22 - 23

F: < 22

For those of you who have received Fx, please contact Pierre. Your task for obtaining a passing grade will involve the topics on which you performed poorly on the exam (different for different individuals). Note that scores close to a passing grade have been re-checked after the initial grading. Thus, for any complaints on such scores you should make sure that you can make a strong case for why you should be awarded more points.

From the statistics (see next section) it is clear that problems 4 and 5 were more difficult than expected. In particular, the most common mistakes on both problems concerned basic probability, such as computing expectations and covariances, rules for conditioning and similar.

The other somewhat unexpected outcome was the poor performance on problem 3b. This was mainly due to many of you not formulating your intuitive thoughts on the problem in proper mathematical terms. Most of you, especially students who passed the exam, seemed to have an idea of how the solution should go, but did not / were not able to formulate it in a precise manner.

Total number of students that took the exam:

`## [1] 120`

Number of students that passed:

`## [1] 76`

Number of students that failed:

`## [1] 44`

Histogram for individual final scores (including bonus points):

A box plot for the score on the five problems:

Summary statistics for all students:

```
## Prob 1 Prob 2 Prob 3 Prob 4
## Min. : 0.000 Min. : 0.000 Min. : 0.000 Min. : 0.000
## 1st Qu.: 5.000 1st Qu.: 4.750 1st Qu.: 3.000 1st Qu.: 0.000
## Median : 7.000 Median : 7.000 Median : 3.000 Median : 2.000
## Mean : 6.571 Mean : 6.067 Mean : 3.475 Mean : 2.504
## 3rd Qu.: 9.000 3rd Qu.: 9.000 3rd Qu.: 3.000 3rd Qu.: 4.000
## Max. :10.000 Max. :10.000 Max. :10.000 Max. :10.000
## Prob 5 Bonus Exam Sum
## Min. : 0 Min. :0.000 Min. : 0.00 Min. : 0.00
## 1st Qu.: 2 1st Qu.:2.000 1st Qu.:17.00 1st Qu.:20.00
## Median : 4 Median :4.000 Median :23.00 Median :26.50
## Mean : 4 Mean :3.158 Mean :22.62 Mean :25.77
## 3rd Qu.: 6 3rd Qu.:5.000 3rd Qu.:28.25 3rd Qu.:32.00
## Max. :10 Max. :5.000 Max. :47.00 Max. :52.00
```

Summary statistics for students who passed the exam:

```
## Prob 1 Prob 2 Prob 3 Prob 4
## Min. : 3.00 Min. : 0.00 Min. : 0.000 Min. : 0.000
## 1st Qu.: 7.00 1st Qu.: 7.00 1st Qu.: 3.000 1st Qu.: 2.000
## Median : 8.00 Median : 9.00 Median : 3.000 Median : 3.000
## Mean : 8.02 Mean : 7.75 Mean : 3.868 Mean : 3.493
## 3rd Qu.: 9.25 3rd Qu.: 9.00 3rd Qu.: 3.000 3rd Qu.: 5.000
## Max. :10.00 Max. :10.00 Max. :10.000 Max. :10.000
## Prob 5 Bonus Exam Sum
## Min. : 0.000 Min. :0.000 Min. :19.00 Min. :24.00
## 1st Qu.: 4.000 1st Qu.:2.000 1st Qu.:24.00 1st Qu.:27.00
## Median : 5.000 Median :5.000 Median :27.00 Median :30.00
## Mean : 5.092 Mean :3.632 Mean :28.22 Mean :31.86
## 3rd Qu.: 7.000 3rd Qu.:5.000 3rd Qu.:33.00 3rd Qu.:36.25
## Max. :10.000 Max. :5.000 Max. :47.00 Max. :52.00
```

Summary statistics for students who failed the exam:

```
## Prob 1 Prob 2 Prob 3 Prob 4
## Min. : 0.000 Min. : 0.000 Min. : 0.000 Min. :0.0000
## 1st Qu.: 1.750 1st Qu.: 0.000 1st Qu.: 2.500 1st Qu.:0.0000
## Median : 5.000 Median : 1.000 Median : 3.000 Median :0.0000
## Mean : 4.068 Mean : 3.159 Mean : 2.795 Mean :0.7955
## 3rd Qu.: 6.000 3rd Qu.: 6.000 3rd Qu.: 3.000 3rd Qu.:1.0000
## Max. :10.000 Max. :10.000 Max. :10.000 Max. :4.0000
## Prob 5 Bonus Exam Sum
## Min. : 0.000 Min. :0.000 Min. : 0.00 Min. : 0.00
## 1st Qu.: 0.000 1st Qu.:0.000 1st Qu.: 9.00 1st Qu.:11.00
## Median : 1.500 Median :2.000 Median :14.00 Median :17.00
## Mean : 2.114 Mean :2.341 Mean :12.93 Mean :15.27
## 3rd Qu.: 3.250 3rd Qu.:5.000 3rd Qu.:18.00 3rd Qu.:21.00
## Max. :10.000 Max. :5.000 Max. :22.00 Max. :23.00
```

The grade distribution was as follows:

`## A: 9.166667 %`

`## B: 11.66667 %`

`## C: 7.5 %`

`## D: 21.66667 %`

`## E: 13.33333 %`

`## Fx: 7.5 %`

`## F: 29.16667 %`