Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

We consider a multiple-choice test with exactly three yes/no-questions. The following rating scheme is given: correct answer +1 point, wrong/missing answer -0.5 points - a negative result is NOT rounded up to 0. Now assume a student who is trying to guess all answers and who is ticking at most one possible answer. The probability that the student leaves one question is unchecked is 1/5 while the probability of ticking one of the two other answers is 2/5. Let A denote a random variable representing the points one student has collected. [...]

Determine parameters $a,b\in\mathbb{R}$ for $A=aX+b$ with $X\sim \text{Bin}(3, 2/5)$ and determine $\mathbb{E}[A]$.

(horribly translated assignment)

The only things I know right now, is that with $$\mathbb{E}[A]=\mathbb{E}[aX+b]=a\cdot\mathbb{E}[X]+b=a\cdot\left(3\cdot\frac{2}{5}\right)+b$$ I could do some magic, but I don't exactly know how to include the possible results (-1.5, 0, 1.5, 3) into the parameters $a$ and $b$.

Could someone give me some hints how to solve this?

share|cite|improve this question
Hint: The student receives one point for a correct answer and $-1/2$ for an incorrect or blank answer and he gets $X$ right (getting $+1$ for each), while $3-X$ are marked incorrect or left blank (getting $-1/2$ for each). Hence, $A$, the student's score on the exam is given by .... – Dilip Sarwate May 19 '12 at 21:42
@DilipSarwate: Thats a really good hint (almost a full solution!). Shame on me why I haven't understood it before. This should result in $\mathbb{E}[A]=0.3$, shouldn't it? – Christian Ivicevic May 19 '12 at 21:49
up vote 2 down vote accepted

You want a linear transformation to change $(0,1,2,3)$ into $(-1.5,0,1.5,3)$.

The gaps are $1$ in the first case and $1.5$ in the second. So $a=\frac{1.5}{1} =1.5$.

Multiplying by $1.5$ changes $(0,1,2,3)$ into $(0,1.5,3,4.5)$. To get from $(0,1.5,3,4.5)$ to $(-1.5,0,1.5,3)$ you have to subtract $1.5$ from each term. So $b=-1.5$.

share|cite|improve this answer

Hint: Let $X_i=1$ if Question $i$ is answered correctly, and $X_i=0$ otherwise. Then the number of marks $M_i$ earned on Question $i$ is $1.5X_i -0.5$. (Check that this is correct in both the case $X_i=1$ and $X_i=0$.)

The total mark is $M=M_1+M_2+M_3$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.