Problem 91 - An Introduction to Mathematical Statistics and Its Applications

Question

Two distinct integers are chosen at random from the first five positive integers. Compute the expected value of the absolute value of the difference of the two numbers.

Step-by-Step Solution

Answer

The expected value of the absolute value of the difference between any two distinct integers chosen randomly from the first five positive integers is 2.4.

Step 1: Understand Expected value

The expected value in statistics is essentially the mean or average of a set of values. It represents the value that one would 'expect' to obtain after multiple trials or observations.

Step 2: Define the Set

Define the set of the first five positive integers, which is \{1, 2, 3, 4, 5\} and note that there are \(5 \times 4 =20 \) pairs of distinct integers that can be chosen.

Step 3: Calculate the Absolute Difference

Calculate the absolute differences of all pairs. The differences are \[0, 1, 2, 3, 4, 1, 2, 3, 4, 2, 3, 4, 3, 4, 4, 2, 3, 3, 1, 2\] obtained by calculating \( |x-y| \) for each possibles pairs (x,y) of integers in the set.

Step 4: Calculate the Expected Value

Now sum all the absolute differences and divide by the total number of pairs to find the expected value. The sum of the differences is 48. Therefore, the expected value of the absolute differences is \(\frac{48}{20} = 2.4\).

Key Concepts

Statistics

Statistics is a field of study about gathering, analyzing, and interpreting data to make informed decisions or predictions. One important concept in statistics is the **expected value**.
Think of the expected value as the "long-run average" or the mean you might expect to see if you were to repeat a random experiment many times.
In our exercise, we are computing expected values for the absolute difference between pairs of integers.
To calculate this, you need to follow these steps:

• List your possible outcomes. In this case, the set \( \{1, 2, 3, 4, 5\} \), provides us possible pairs of integers.
• Determine the result (i.e., absolute difference) for each unique pair.
• Take the sum of all these absolute differences and divide by the number of pairs to find the "average" result.

By finding the average of these outcomes, you learn what to "expect". So, we expect the average absolute difference of two integers drawn from our set randomly to be 2.4. This concept is a powerful tool in decision-making processes where probabilities are involved clear as letting you know the most likely outcome.

Discrete Mathematics

Discrete mathematics deals with distinct and separate values, often counting and arrangement of objects. In this exercise, the numbers we are working with (from 1 to 5) fall into the category of discrete values.
When computing something like the expected value of the absolute difference between pairs of discrete numbers, it's crucial to understand basic principles of discrete math:

• Each number or result is distinct and their outcomes can be listed or counted individually and in a finite manner, which is exactly what we do by listing all possible differences.
• Understanding combinations and permutations is key, as we saw when calculating the total number of pairs \( \( \binom{5}{2} = 10\) \), but without repetition and consistent with the problem's context.

This makes discrete mathematics highly relevant when dealing with probability where distinct possible outcomes and their arrangements or combinations are a focus.

Probability Theory

Probability theory is the branch of mathematics concerned with the analysis of random phenomena. The central idea is to quantify uncertain events and their outcomes.
In our exercise, each pair of numbers is a random choice from a finite set, and we want to calculate the expected value of their absolute difference.
The steps involve:

• Listing potential outcomes which are the set of all absolute differences obtained by taking two distinct numbers from \( \{1, 2, 3, 4, 5\} \).
• Determining the probability of each outcome if needed for more complex problems; here, each pair is chosen with equal likelihood.
• Calculating averages to find out what we should typically "expect" in a random system.

By handling these elements, probability theory enables us to predict with accuracy, allowing us to understand the typical behavior or outcome of random events, such as finding the average absolute difference in our integer set example.