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Suppose that the distribution of time at a single job is a gamma distribution with a mean of $5$ years and a standard deviation of $2$ years, and suppose that the times at successive jobs are i.i.d. random variables. Every year the person fills out the survey. What are the approximate average answers to the following questions:

a. How long have been working at your current job?

b. How long did you work at your last job?

My attempt: (b) From the given information, the average time spent at the last job should follow gamma distribution with mean $5$ years, so the answer should be $\fbox{5}$ years?

(a) Let $Y_i =$ the time spent at current job. Then it also follows gamma distribution, so $E(Y_i) = 5$ years. Let $T_i = $ transition time from the last job to the current job are random variable, but we are not given $E(T_i)$. How could we solve this part?

My question: I don't see the difference between the two questions, as in both cases, the question seems not to account for the transition time. Also, we are not given the average transition time between successive jobs. Could someone please help with the correct interpretation?

ghjk
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  • Your answer to part (a) assumes that you're at the end of your current job. – T.J. Gaffney Nov 16 '16 at 00:16
  • @Gaffney: I was thinking of a renewal process, where $N(t) =$ the time between successive jobs. So part (b) is asking $N(t)/t$ as $t\rightarrow \infty$. By using the limit theorem, that fraction equals $\frac{1}{5}$. Do you think this is correct? I don't see how to define a renewal process in part (a) btw:p – ghjk Nov 16 '16 at 00:54
  • So for part (a), my current thought is follows: let $X_n = $ average time spent on current job answered between successive surveys. This is the renewal process, so $E(X_n) = \frac{E(X_n)}{2} + \frac{Var(X_n)}{2E(X_n)} = \frac{5}{2} + \frac{4}{2\times 5} = 2.9$ years. Is this correct? – ghjk Nov 16 '16 at 03:20
  • Could anyone please give some thoughts to my attempted solution in the comment, or to the original problem? – ghjk Nov 17 '16 at 17:37

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