How Is the Conditional PMF Calculated for Car Arrivals at a Toll Station?

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The discussion focuses on calculating the conditional probability mass function (pmf) for car arrivals at a toll station, modeled as a Poisson process with a rate of 5 cars per minute. Given that five cars arrived in a three-minute interval, the conditional pmf for the number of cars arriving in the first minute can be derived using properties of the Poisson process. Specifically, the distribution of arrivals is uniform across the interval, allowing for the application of the multinomial distribution to determine the conditional probabilities. This highlights the unique characteristics of the Poisson process, often referred to as the "most random" point process. Understanding these principles is crucial for accurately modeling and predicting arrival patterns.
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Suppose that cars traveling at night on a freeway arrive at a toll station according to a Poisson process with rate alpha @=5 per minute. If five cars arrived in [0,3] (that is, during a three minute period starting at midnight), what is the conditional pmf of the number of cars that arrived in [0,1] (during the first minute of the period)?
 
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What do you mean by pmf? Probability mass function?

Anyways, You condition on the even that up t=3 five cars arrived. what do you know about the conditional distribution of the arrival times of these five cars?

Hint: It's a famous property of the Poisson process because of which it is sometimes referred to as the "most random" point process of all.
 

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