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Yousha takes the train every day from Jangpura, where he lives, to go to his school at INA train station. Jangpura is on the Violet line, while INA is on the yellow line and the interchange is at Lajpat Nagar station. The waiting times in Jangpura and Lajpat Nagar are independent random variables following the exponential law with parameter λ. Find the probability law of the total waiting time in the two stations.

Any help will be appreciated. I have no idea where to take it from! I thought I was familiar with combinatorics and probabilities but this is completely out of my knowledge base!!

Pradeep Suny
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    In probability theory language the exercise asks you what the distribution is of $X+Y$ when both are Exponential$(\lambda)$ distributed and independent from one another. Do you know what a convolution is? https://en.wikipedia.org/wiki/Convolution_of_probability_distributions. – Marc Sep 30 '19 at 13:49
  • @Marc not really :( – Pradeep Suny Sep 30 '19 at 13:59
  • This isn't combinatorics or discrete probabilities. It deals with continuous probability distributions. Have you studied that yet? – saulspatz Sep 30 '19 at 14:52
  • @Marc is absolutely correct: for the total time to be "x" for example, the individual times must add up to x. Than can happen in many ways: the two times can be x/2 each or 0.1 x and 0.9 x and so on. Add (integrate) up all those values to find the total change the waiting time is x. –  Sep 30 '19 at 16:40
  • @saulspatz: I am not a student; I am almost 40 and trying to refresh my knowledge in probabilities and distributions because I frequently run onto statistics problems in my job. I have studied elementary probabilities etc in my Engineering bachelor. – Pradeep Suny Sep 30 '19 at 18:21
  • My first instinct is to look it up at wikipedia but surprisingly some of the formulas there assume $\lambda_1 \neq \lambda_2$. You will have to modify the algebraic manipulations a bit, but the convolution is explicitly shown. – antkam Sep 30 '19 at 18:39

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In probability theory language the exercise asks you what the distribution is of $X+Y$ when both are Exponential$(\lambda)$ distributed and independent from one another. Define $Z = X+Y$, then by the convolution formula we have \begin{align} f_Z(z) &= \int_{-\infty}^\infty f_X(t)f_Y(z-t) dt = \int_{0}^z f_X(t)f_Y(z-t) dt \\ &= \int_{0}^{z}\lambda e^{-\lambda t} \lambda e^{-\lambda (z-t)}dt = \int_{0}^{z}\lambda^2e^{-\lambda z} dt = \lambda^2ze^{-\lambda z}. \end{align} This is the pdf of the Gamma distribution with shape equal to two and rate equal to $\lambda$. See https://en.wikipedia.org/wiki/Gamma_distribution.

Marc
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