Problem 39 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose a fair die is tossed three times. Let \(X\) be the largest of the three faces that appear. Find \(p_{X}(k)\).

Step-by-Step Solution

Answer

The probability mass function \(p_{X}(k)\) is \(p_{X}(k) = \frac {k^{3} - (k-1)^{3}}{6^3}\).

Step 1: Identify total possible outcomes

When rolling a six-sided die three times, the total number of possible outcomes is \({6^3} = 216\). This is because each roll can result in one of six outcomes (1 to 6 inclusive).

Step 2: Calculate favourable outcomes for each \(k\)

To find \(p_{X}(k)\), we need to find the number of ways in which \(k\) can be the maximum outcome in three rolls. If \(k\) is the largest number, any rolls can be any number from 1 to \(k\), and there are \(k^{3}\) ways. Subtract from this the ways that the numbers—1 to \(k-1\) can appear, we get: \(k^{3} - (k-1)^{3}\).

Step 3: Find \(p_{X}(k)\)

Substitute the favourable outcomes and the total outcomes into the probability formula to calculate \(p_{X}(k)\): The probability mass function \(p_{X}(k)\) can be given as: \(p_{X}(k) = \frac {k^{3} - (k-1)^{3}}{6^3}\).

Key Concepts

Probability Mass Function

A probability mass function (PMF) is a vital tool in probability theory and statistics. It helps you understand the likelihood of different outcomes for discrete random variables. Think of a PMF as a mapping from each possible value of a random variable to the probability that it will occur.

For a given discrete random variable, like the largest face showing on a die after three rolls, a PMF assigns probabilities to each potential maximum value. This function needs to satisfy two primary conditions:

• Each probability value assigned must be between 0 and 1.
• The sum of all probabilities in the PMF must equal 1.

When rolling a fair die three times, the possible values for the largest face, denoted as random variable \( X \), are integers from 1 to 6. The PMF, \( p_{X}(k) \), then gives you the probability of \( X \) taking each value \( k \). By understanding the PMF, you grasp how to distribute these probabilities according to the values that \( X \) can take, ensuring all outcomes are covered.

Discrete Random Variables

A discrete random variable is a type of variable that can take on a finite or countably infinite number of distinct values. This is in contrast to continuous random variables which can take an infinite number of possible values over a continuum.

In the context of our exercise, the random variable \( X \) represents the largest face value obtained when rolling a die three times. The randomness comes from the unpredictable nature of each die roll, and each outcome is distinct and countable, which makes \( X \) a classic example of a discrete random variable.

The values that a discrete random variable can take are often mapped in a probability mass function (PMF). This relationship allows you to see at a glance the probability associated with each potential outcome, providing a clear picture of how the random variable behaves. Using PMFs for discrete random variables helps in simplifying complex probability problems into manageable calculations.

Combinatorial Analysis

Combinatorial analysis is a mathematical technique used to count or enumerate various possible outcomes and configurations. It is the backbone of computing probabilities in scenarios where outcomes are not easily listed individually, like our dice problem.

In our exercise, combinatorial analysis helps us determine how many ways each value of \( k \), the largest face, can appear when rolling a die multiple times. This involves counting the number of successful outcomes and comparing it to the total possible outcomes. Here's how it works step-by-step:

• First, identify the total number of possible outcomes when rolling the die three times, which is \( 6^3 = 216 \).
• Then, calculate the favorable outcomes for each maximum value \( k \). For \( k \) to be the maximum, each roll could have been any number from 1 to \( k \). This gives \( k^3 \) possible sequences.
• Subtract the sequences where the maximum is less than \( k \), which totals \((k-1)^3\).
• The difference, \( k^3 - (k-1)^3 \), gives the favorable outcomes for \( k \), and dividing by the total outcomes reaches the desired probability.

Thus, combinatorial analysis helps to systematically break down and solve problems involving probabilities, yielding precise and clear results.