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Two tram lines operate independently of each other on a certain route, both $t$ minutes apart. Person $A$ comes to the stop and chooses the train that comes first. Determine the density function of his waiting time and the expected value.

If let $W_1$ and $W_2$ denote the waiting times for both trains such that they're uniformly distributed in $[0,t]$ the probability that $W_1 > w$ is $P(W_1 > w) = \frac{t-w}{t}$ and similarly for $W_2$. The probability that both are greater than $w$ is then $$\left( \frac{t-w}{t}\right)^2.$$ So is the probability of the minimum waiting time say $W$ the same as this $\left( \frac{t-w}{t}\right)^2$ and if so can I use this to derive the density function?

Iman
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1 Answers1

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If I understand the phrasing of the question correctly, there are two tram lines B and C. The time between each tram for the lines separately is always $t$. However, there is some offset between the two lines that is randomly determined -- say, the trains of line C always arrive $pt$ after the trains of line B, where $p\in [0,1]$. I assume that p is considered to be drawn uniformly at random. Finally, we assume that person A arrives at a time $q$, and the question is how long they have to wait.

Without loss of generality, we can shift the time coordinate so that person A arrives at time $0$. Then, the waiting times can be considered to be two random variables that are uniformly distributed on $[0,1]$, and since we take the first tram, we are interested in the minimum. Hence, we are in the setting of the following question: Expectation of Minimum of $n$ i.i.d. uniform random variables.

If my interpretation is wrong, please correct me! Otherwise, it appears that the answer can be found in the linked question.

FinleyMarsh
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