### General information

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.

### Statistics

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 %