I am trying to solve the following question:

A class has 20 students, after the first month, the professor has figured out that arrival times T1, T2, ... ,T20 of students are independent and exponentially distributed random variables with parameters λ1,λ2,...,λ20 respectively.

a. What is the distribution of R.V M that describes the arrival time of first student?

b. If arrival rate of each student is the same, what is the expected value of time until the last student has arrived to attend the class?

As for the a, I believes as we are interested in finding the time of first arrival it should be Exponential distribution with parameter λ1.

But regarding b, I am quite confuse, intuitively I believe it should be equal to the arrival rate of students(as it is equal for all students), but I am not sure how to express it in a better manner?


  • 1
    $\begingroup$ I suspect "first student" means earliest of arrive (so an order statistic) rather than student with index number $1$ $\endgroup$ – Henry Nov 9 '15 at 8:26

a. You have to calculate is the distribution of the random variable $\min \{ T_1,\dots,T_{n} \}$, where $T_i \sim \textrm{Exp}(\lambda_i)$. It is well known that $$ \min \{ T_1,\dots,T_{n} \} \stackrel{d}{=} \textrm{Exp}(\lambda_1+\dots+\lambda_{n}), $$ you can easily prove this for a general $n$ and $\lambda$'s using CDF (see for instance the proof for the case $n=2$ in this other thread).

b. Assume that $T_i \sim \textrm{Exp}(\lambda)$ for every $i$. Using the memoryless property of exponential random variables, one can show that $$\max \{ T_1,\dots,T_{n} \} \stackrel{d}{=} \sum_{k=1}^n X_k,$$ where $X_k \sim \textrm{Exp}(\lambda \cdot k)$, so in particular $\mathbb E \max \{ T_1,\dots,T_{n} \} = \sum_{i =k}^n \frac{1}{\lambda \cdot k }$.


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