- #1
Mehmood_Yasir
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Homework Statement
Pedestrians approach to a signal at the crossing in a Poisson manner with arrival rate ##\lambda## arrivals per minute. The first pedestrian arriving the signal starts a timer ##T## then waits for time ##T##. A light is flashed after time T, and all waiting pedestrians who arrive within time duration ##T## must cross.
What is the probability that a randomly arriving padestrian has crossed the crossing in a group of ##k+1## padestrians? Here is the group is called ##k+1## because first one starts ##T## and ##k## more padestrian arrive within time ##T##.
The answer given is ##\frac{(K+1) {(\lambda * T)}^k e^{-\lambda T} } {k! (1+\lambda T)}##
I could not understand this answer. Can someone kindly explain me this answer.
Homework Equations
Poisson formula for general ##k## arrivals in time ##T##,
##P_k= \frac{{(\lambda * T)}^k e^{-\lambda T} } {k! }##
The Attempt at a Solution
Since the total padestrian which has crossed the crossing after light is flashed = ##k+1##
We know that for a Poisson process, the probability of ##k## arrivals in a given time interval ##T## is ## \frac{{(\lambda * T)}^k e^{-\lambda T} } {k! }##.
Probability that ##k+1## has crossed the crossing is ## \frac{{(\lambda T)}^{k+1} e^{-\lambda T} } {(k+1)! }##
What is the probability that a randomly arriving padestrian has crossed the crossing in a group of ##(k+1)## ??