Probability distribution of first arrival time in Poisson Process

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SUMMARY

The discussion centers on the probability distribution of the first arrival time in a Poisson process, specifically addressing the calculation of the waiting time's probability density function. The probability for the waiting time to observe the first arrival is given by P(T1>t)=exp(-lambda*t). The probability density function for the waiting time itself is expressed as P(T1=t)=λe^(-λt), where λ represents the rate of the Poisson process. This highlights the continuous nature of the distribution, confirming that the probability of T1 equaling a specific value is zero.

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phyalan
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According to wiki:
http://en.wikipedia.org/wiki/Poisson_process

The probability for the waiting time to observe first arrival in a Poisson process P(T1>t)=exp(-lambda*t)
But what is the Probability Distribution P(T1=t) of the waiting time itself? How to calculate that?
 
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Since the waiting time has a continuous distribution, the probability of T1 having a particular value = 0. I suppose you would want the density function, λe-λt.
 

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