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 that will happen on any of these highways today is \(1.5\) accidents.

Step 1: Identify the Poisson parameters

The given parameters for Poisson random variable representing daily accidents on highways are .3, .5, and .7.

Step 2: Calculate expected accidents for each highway

The expected value of a Poisson random variable is equal to its parameter. Therefore, for the three highways, their expected accidents are: - Highway 1: E(X_1) = .3 - Highway 2: E(X_2) = .5 - Highway 3: E(X_3) = .7

Step 3: Calculate overall expected accidents

To find the total expected accidents, we add the expected accidents of all three highways: Expected Accidents = E(X_1) + E(X_2) + E(X_3) Expected Accidents = .3 + .5 + .7 Expected Accidents = 1.5 So, the expected number of accidents that will happen on any of these highways today is 1.5 accidents.

Key Concepts

Expected Value

The expected value, often represented as \( E(X) \), is a fundamental concept in probability theory and statistics, serving as the "mean" or "average" of a random variable. In simple terms, it indicates what we would expect to happen on average across many, many trials. Think of it as a weighted average of all possible outcomes, where each outcome is weighted according to its probability of occurrence.

For a Poisson distribution, which is commonly used to model random events like accidents or arrivals that occur independently over a fixed period of time, the expected value equals the distribution's parameter, \( \lambda \).

Key points about expected value include:

• It gives a central value for random variables.
• In Poisson distribution, \( E(X) = \lambda \).
• It is used to predict long-term behavior over many repetitions of an experiment or process.

In the exercise, each highway has its own "lambda" which dictates the expected number of accidents, and the totals were summed to find the combined expectation across all highways.

Random Variable

A random variable is a core concept in probability theory, where it acts like a function assigning numerical values to each possible outcome in a sample space. There are two main types of random variables: discrete and continuous. In the context of our exercise, we're dealing with discrete random variables since we are counting the number of accidents, which can only be whole numbers.

Understanding what random variables represent:

• They link outcomes of a random process to numerical values.
• Discreteness, as in this case, implies outcomes like accidents per day are countable (0, 1, 2, etc.).
• Continuous random variables, on the other hand, could take on any value within a range, like measurements of time or distance.

Here, Poisson random variables are used to model processes with a known average rate of occurrence, where the number of events in one interval is independent of events in other intervals.

Probability Theory

Probability theory is a branch of mathematics concerned with the analysis of random phenomena. It provides the mathematical foundations for statistics and underpins the calculation of expected values, variances, and other statistical properties. It's the backbone for understanding concepts like Poisson distributions and their applications in real-world scenarios, such as predicting traffic accidents on highways.

A few crucial aspects of probability theory include:

• Defining events and assigning probabilities to them, whereby the probability is a number between 0 and 1.
• Understanding independent events and how their probabilities can combine (e.g., the sum of probabilities of all possible outcomes equals 1).
• Using models like Poisson distribution for specific types of random processes.

In the exercise discussed, probability theory allows us to use the parameters provided (\( \lambda \'s \)) to find how likely different kinds of accident scenarios are on each highway, as well as their combined rate, giving us the framework to compute expected outcomes.