Problem 25 - A First Course in Probability

Question

Suppose that \(10^{6}\) people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over \(\left(0,10^{6}\right) .\) Let \(N\) denote the number that arrive in the first hour. Find an approximation for \(P\\{N=i\\}.\)

Step-by-Step Solution

Answer

The approximation for \(P(N=i)\) is given by the formula: \[P(N=i) \approx e^{-1}\frac{1^i}{i!}\]

Step 1: Determine the average number of arrivals in the first hour

To determine the average number of arrivals in the first hour, we first observe that there are \(10^6\) people arriving over a total of \(10^6\) hours. The uniform distribution implies that the average number of arrivals per hour is the same. Therefore, the average number of arrivals in the first hour is: \[\lambda = \frac{10^6}{10^6} = 1\]

Step 2: Find the Poisson distribution formula

The Poisson distribution is given by the formula: \[P(X=k) = e^{-\lambda}\frac{\lambda^k}{k!}\] In our case, we want to approximate the probability \(P(N=i)\) for some value \(i\). Since \(\lambda = 1\), we can substitute this value into the Poisson distribution formula: \[P(N=i) \approx e^{-1}\frac{1^i}{i!}\]

Step 3: Compute the approximation for P(N=i)

Now that we have the approximation formula for \(P(N=i)\), we can plug in any value of \(i\) to calculate the corresponding probability. For example, if we want to approximate the probability that exactly 2 people will arrive in the first hour, we can plug in \(i = 2\): \[P(N=2) \approx e^{-1}\frac{1^2}{2!} = \frac{1}{2e}\] In general, the approximation for \(P(N=i)\) can be given by: \[P(N=i) \approx e^{-1}\frac{1^i}{i!}\]

Key Concepts

Random Variables

In the realm of statistics and probability, a random variable is a quantifiable feature that assigns numerical values to the outcomes of a random phenomenon. When you repeat an experiment under identical conditions, you might get different results because of the randomness involved. For example, tossing a coin can result in either heads or tails—a random variable can represent this with numbers, like 1 for heads and 0 for tails.

In our exercise, the arrival time of each individual is a random variable since we cannot predict it exactly. The fact that they are independent means the arrival time of one person does not affect anyone else's arrival time—a foundational aspect for using the uniform distribution and Poisson distribution to approximate probabilities.

Uniform Distribution

The uniform distribution is one of the simplest types of probability distributions. It's distinctive for the fact that every outcome in a range has an equal chance of occurring. Picture a perfectly balanced dice; when you roll it, each face—numbered 1 through 6—is equally likely to come up. That's the essence of uniform distribution.

The exercise presents a scenario where the arrival times are uniformly distributed over a large interval, \(0,10^6\). This implies every moment within that interval is equally probable for a person to arrive. Because of this equal probability, the expected number of people arriving during any particular hour is consistent across all hours, hence the use of uniform distribution as a model.

Probability Approximation

When working with probabilities, sometimes exact calculations are complex or even impossible. This is where probability approximation comes into play, providing a simplified model that is close enough to the true probabilities for practical use. The Poisson distribution is often employed as an approximation in cases where events are rare or the number of trials is large, as in the case of the service station problem.

The Poisson distribution approximates the probability of a given number of events occurring in a fixed interval, assuming these events happen at a constant rate and independently of the time since the last event. In the service station example, the Poisson distribution simplifies calculating the likelihood of a certain number of arrivals in the first hour, based on a constant average rate of \(\lambda = 1\). This powerful approximation tool allows one to estimate probabilities quickly and with reasonable accuracy, particularly for large-scale scenarios or when dealing with random, infrequent events.