Problem 168 - An Introduction to Mathematical Statistics and Its Applications

Question

Two fair dice are tossed. Let \(X\) denote the number appearing on the first die and \(Y\) the number on the second. Show that \(X\) and \(Y\) are independent.

Step-by-Step Solution

Answer

Yes, \(X\) and \(Y\) are independent because their joint probability distribution equals the product of their individualprobability distributions.

Step 1: Define individual probabilities

Each die has 6 outcomes, and each outcome is equally likely. Therefore, the probability of any given outcome on one die is \( \frac{1}{6} \). Hence, \( P(X = x) = P(Y = y) = \frac{1}{6} \) for \( x, y \) in {1, 2, 3, 4, 5, 6}.

Step 2: Define joint probability

The joint probability of two events \(X\) and \(Y\) is defined as \( P(X = x, Y = y) \). Since we have two dice and each die has 6 outcomes, there are a total of \(6 * 6 = 36\) possible outcomes. Therefore, the joint probability \( P(X = x, Y = y) = \frac{1}{36} \) for \( x, y \) in {1, 2, 3, 4, 5, 6}.

Step 3: Prove Independence

We need to show that \( P(X = x, Y = y) = P(X = x) * P(Y = y) \). Substituting the known probabilities, we get \( \frac{1}{36} = \frac{1}{6} * \frac{1}{6} \), which simplifies to \( \frac{1}{36} = \frac{1}{36} \). Thus, we can conclude that \(X\) and \(Y\) are independent.

Key Concepts

Independent Events

In probability theory, independent events are events whose outcomes do not affect each other. Imagine two fair dice being thrown. The result of one die does not change or influence the outcome of the other die. This makes them independent.

For two events to be independent, the probability of both occurring together must equal the product of their individual probabilities. Mathematically, two events \( A \) and \( B \) are independent if:

• \( P(A \cap B) = P(A) \times P(B) \)

This condition essentially checks whether the joint probability (probability of both events happening) of any two events is equal to the multiplication of their separate probabilities. If this holds true, the events are independent.

Joint Probability

Joint probability refers to the probability of two events occurring at the same time. In our exercise, it's the probability that the first die rolls a certain number \( X \), and the second die rolls a certain number \( Y \).

When dealing with dice, each outcome is equally likely. So the joint probability of any two specific outcomes is determined by the product of individual probabilities. For two dice, both having 6 outcomes, the joint probability is given by:

• \( P(X = x, Y = y) = \frac{1}{36} \)

Understanding joint probability is crucial when working with multiple random variables. It helps in analyzing how two or more events might intersect or occur together.

Fair Dice

A fair die is a theoretical dice that has an equal chance of landing on any of its faces. This means each side (or outcome) is equally probable. For a standard six-sided die, each face has a probability of:

• \( \frac{1}{6} \)

Fairness ensures that each roll of a die is unbiased and random. This is important in probability theory, as it allows us to calculate probabilities accurately.

In scenarios involving fair dice, like our exercise, knowing this fact allows us to determine the probabilities of combined events, given that each possible result is equally likely.

Random Variables

In probability theory, a random variable represents a numerical outcome of a random event. It is a variable whose possible values are numerical outcomes of a random phenomenon. In our exercise, \( X \) and \( Y \) are random variables representing the numbers on two dice.

Random variables can be discrete, taking on a finite set of values (like a die roll), or continuous, taking on a range of values. In the case of dice:

• \( X \) and \( Y \) are discrete random variables.
• Each can take on any integer value from 1 to 6.

With random variables, we can assign probabilities to various outcomes, enabling the study and prediction of different scenarios in probability theory.