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making a practice exam I had to make the following problem which I couldn't solve unfortunately...

Problem

In the springtime, a student has $N$ days to find a summer job for one month. Each day, the student is offered a job with total salary drawn from a exponential distribution with mean $R > 0$. The offers are independent. The costs of applying for jobs is $c < R$; the student pays these costs each time he rejects an offer. The offer on day $N$ is accepted in all cases. He cannot revisit a rejected offer. When should the student accept an offer?

I hope someone can help me..

kees den otter
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2 Answers2

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The last day's offer has expected pay $R$.

Let's suppose the expected value if you start the $k$th day is $R_k$, so $R_N=R$.

Therefore the $k-1$th day's offer should be accepted if and only if its offer is greater than $R_k-c$, since you could instead choose to pay $c$ and then take your chance on the $k$th day.

If you start the $k-1$th day, the probability of accepting that day's offer is $\exp\left(-\frac{R_k-c}{R}\right)$ and conditioned on accepting it then (using memorylessness) the expected pay is $R+R_k-c$. So the expected position if you reach the $k-1$th day is $R_{k-1}=(R+R_k-c)\exp\left(-\frac{R_k-c}{R}\right) + (R_k-c)\left(1-\exp\left(-\frac{R_k-c}{R}\right)\right)$, i.e. $R_{k-1}= R\exp\left(-\frac{R_k-c}{R}\right) +R_k -c$.

Henry
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  • @Thank you, could you please explain how I can solve the recursive formula you showed me? – Nedellyzer Nov 10 '13 at 13:27
  • I would do it numerically – Henry Nov 10 '13 at 13:32
  • But I have to solve it for general $N$ – Nedellyzer Nov 10 '13 at 13:33
  • use some sofware,like matlab and choose different $N$,and compare results – dato datuashvili Nov 10 '13 at 13:38
  • This method works for general $N$, just not in a closed form. The answer is "accept the $k−1$th day's offer if and only if its offer is greater than $R_k−c$; if you reach the last day, accept that offer". – Henry Nov 10 '13 at 13:39
  • @Henry I see what you mean, but you cannot revisit the $k-1$th offer if you know the $k$th offer, am I right? – Nedellyzer Nov 10 '13 at 13:43
  • @Sjoerd That is correct. But the values of $R_k$ do not depend on the actual offers. They are just numbers depending on $R$ and $c$ and can all be calculated in advance of the whole process. You then accept an offer on a particular day if it is high enough, and go on to the next day if not. – Henry Nov 10 '13 at 13:48
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we know that exponential distribution has following form exponential

now if student reject offers he paid $c$ right?so if he rejected $x$ times,it means cost $x*c$ is it right?so our problem is that we have to determine number of days he should reject so that cost $x*c$ should be minimum as well as salary must be maxsimum,so i think we have to find number of $x$ so that cost must be minimum and salary maximum,we may use Lagrangian multiplier or another method ,

dato datuashvili
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  • I don't see what you mean here, I also don't understand the answer, can you please be more specific? – Nedellyzer Nov 10 '13 at 12:48
  • so your salary is represented by exponential distribution right?imagine that you have rejected job offering $x$ times,cos is $x*c$ right? so now what does mean cost minimum?clearly you can reject on first time,but because for different $x$,salary is different,try to find $x$ which maximize salary and minimize cost,is it clear? – dato datuashvili Nov 10 '13 at 12:59
  • Yes, I know what you mean but the way on how to find this maximum is not clear to me.. – Nedellyzer Nov 10 '13 at 13:04
  • you have to maximize salary-cost – dato datuashvili Nov 10 '13 at 13:16