Problem 83 - A First Course in Probability

Question

There are three highways in the county. The number of daily accidents that occur on these highways are Poisson random variables with respective parameters. \(3, .5,\) and \(.7 .\) Find the expected number of accidents that will happen on any of these highways today.

Step-by-Step Solution

Answer

The expected number of accidents on any of these highways today is 4.2, which is the sum of the expected accidents on each highway: 3 + 0.5 + 0.7.

Step 1: Calculate the Expected Value for Each Highway

For each highway, the expected number of accidents is equal to its parameter. Highway 1: Parameter (λ) = 3 Expected accidents (E) = λ = 3 Highway 2: Parameter (λ) = 0.5 Expected accidents (E) = λ = 0.5 Highway 3: Parameter (λ) = 0.7 Expected accidents (E) = λ = 0.7

Step 2: Calculate the Total Expected Accidents

Next, we sum up the expected accidents on each highway to get the total expected accidents today. Total expected accidents = Expected accidents (Highway 1) + Expected accidents (Highway 2) + Expected accidents (Highway 3) Total expected accidents = 3 + 0.5 + 0.7 = 4.2 Thus, the expected number of accidents on any of these highways today is 4.2.

Key Concepts

Understanding Expected Value

The term 'expected value' is fundamental in the realm of statistics and probability theory. It represents the average outcome if an experiment or random trial was to be repeated many times. For students tackling problems involving expected value, it’s important to understand that this is not about predicting a single outcome, but rather measuring the center of a distribution of outcomes over the long run.

To compute the expected value for a random variable corresponding to a Poisson distribution, as seen in our exercise, one simply uses the parameter \( \lambda \), which indicates the mean number of occurrences over a given interval. The expected value in our scenario helps predict the average number of highway accidents on a given day, assuming that the daily number of accidents follows a Poisson distribution, which is reasonable for rare, independent events happening over a period of time.

Here's how simplicity aids understanding: Imagine you have three jars, each with a different number of marbles that represent the accidents, and the numbers on the jars are the expected values. If you were to reach into the jars and pull out marbles over and over, the numbers on the jars (the expected values) tell you how many marbles you should expect to get on average from each jar.

Random Variables in Probability

Random variables are a foundational concept in probability theory, serving as the bridge between real-world randomness and mathematical abstraction. They assign a numerical value to each outcome of a random process. There are two types of random variables: discrete and continuous. In the context of our exercise, the number of accidents on a highway is a discrete random variable because it counts occurrences, which can only be whole numbers.

Consider the discrete nature of the Poisson process: each accident is a distinct event that can be counted, much like tallying apples in a basket. In any given set of circumstances, knowing the properties of the related random variable helps us describe and predict outcomes. For students, a helpful visual might be to picture random variables as labeled boxes, where each box accumulates data points of similar outcomes from repeated experiments or occurrences, like tossing a die and recording the results.

Fundamentals of Probability Theory

Probability theory is the mathematical backbone that helps us make sense of random phenomena. Using principles from this theory allows us to calculate the likelihood of different outcomes and better understand the nature of uncertainty. In the context of the exercise provided, the Poisson distribution is a probability distribution that is very useful in scenarios where events occur independently and at a constant average rate within a given frame of time or space.

To illustrate, look at the Poisson distribution used to model the highway accidents as a weather forecast but for accidents. It doesn’t tell us exactly when and where an accident will happen; instead, it gives us an idea about how dense the 'accident-event-cloud' is over a highway on a given day. Understanding these concepts is crucial for students, as they make the world of random events much more navigable and less mysterious. It's like having a guide for how often you should bring an umbrella — not to guarantee that you won’t get wet, but to play the odds in your favor.