The number of hours between successive train arrivals at the...

The number of hours between successive train arrivals at the station is uniformly distributed on $(0,1)$. Passengers arrive according to a Poisson process with rate 7 per hour. Suppose a train has just left the station. Let $X$ denote the number of people who get on the next train. Find (a) $E[X]$ (b) $\operatorname{Var}(X)$

Key Concepts

Law of Total Expectation

The law of total expectation, also known as the tower property, states that the expected value of a random variable can be computed by taking the expectation of its conditional expectation over another variable. This is applied in the problem by computing the expectation of the number of passengers boarding the train by first conditioning on the train arrival time and then taking the average over the uniform distribution of that arrival time.

Law of Total Variance

The law of total variance expresses the variance of a random variable as the sum of the expected conditional variance and the variance of the conditional expectation. In the problem, this law is used to calculate the variance of the number of passengers boarding the train by accounting for both the variability within any fixed interval (given by the Poisson process) and the variability introduced by the random train arrival time.

Poisson Process

A Poisson process is a model for a series of events occurring randomly in time or space, where the number of events in any fixed interval follows a Poisson distribution with a parameter that is proportional to the length of the interval. In this problem, passengers arrive according to a Poisson process with a known rate, which allows us to model the count of passengers arriving in a given (random) time period as a Poisson distributed random variable.

Uniform Distribution

The uniform distribution is a continuous probability distribution where every outcome in a given interval has an equal probability of occurring. In the context of the problem, the train arrival time is uniformly distributed on (0,1), meaning that any time duration in that interval is equally likely for the next train's arrival, and properties like its mean (0.5) and variance (1/12) are used in further calculations.

Conditional Expectation

Conditional expectation is the expected value of a random variable given some condition or event. In the problem, the number of passengers boarding the next train is conditioned on the random train arrival time (which is uniformly distributed), and using the conditional expectation simplifies the calculation of the overall expectation by first conditioning on the known interval length.

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