Problem 153 - An Introduction to Mathematical Statistics and Its Applications

Question

. Suppose that \(X\) and \(Y\) are discrete random variables with $$ \begin{aligned} p_{X, Y}(x, y)=& \frac{4 !}{x ! y !(4-x-y) !}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{3}\right)^{y}\left(\frac{1}{6}\right)^{4-x-y}, \\\ & 0 \leq x+y \leq 4 \end{aligned} $$ Find \(p_{X}(x)\) and \(p_{Y}(x)\).

Step-by-Step Solution

Answer

The marginal probabilities \(p_{X}(x)\) and \(p_{Y}(y)\) of the discrete random variables \(X\) and \(Y\) are computed by summing the joint probabilities over all possible values of the other variable. The specific formulas are \(p_{X}(x) = \sum_{y=0}^{4-x} \frac{4 !}{x ! y !(4-x-y)!}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{3}\right)^{y}\left(\frac{1}{6}\right)^{4-x-y}\) and \(p_{Y}(y) = \sum_{x=0}^{4-y} \frac{4 !}{x ! y !(4-x-y)!}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{3}\right)^{y}\left(\frac{1}{6}\right)^{4-x-y}\) respectively.

Step 1: Understand Marginal Probability

The marginal probability of an event \(X\), denoted as \(p_{X}(x)\), is the sum of joint probabilities of \(X\) and \(Y\), keeping \(X\) fixed at a particular value and allowing \(Y\) to vary over all possible values. Similarly, the marginal probability of an event \(Y\) \(p_{Y}(y)\), is computed by summing over all joint probabilities of \(X\) and \(Y\) where \(Y\) is fixed at a certain value and \(X\) is allowed to range over all possible values.

Step 2: Determine \(p_{X}(x)\)

The marginal probability \(p_{X}(x)\) is determined by summing the joint probabilities \(p_{X, Y}(x, y)\) over all possible values of \(Y\). This means that we must compute a sum for \(Y\) ranging from 0 to \(4 - x\). This can be achieved by the following formula: \[ p_{X}(x) = \sum_{y=0}^{4-x} \frac{4 !}{x ! y !(4-x-y)!}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{3}\right)^{y}\left(\frac{1}{6}\right)^{4-x-y}.\]

Step 3: Determine \(p_{Y}(y)\)

Similarly, the marginal probability \(p_{Y}(y)\) is found by summing the joint probabilities \(p_{X, Y}(x, y)\) over all possible values of \(X\). This implies computing a sum for \(X\) ranging from 0 to \(4 - y\). This results in: \[ p_{Y}(y) = \sum_{x=0}^{4-y} \frac{4 !}{x ! y !(4-x-y)!}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{3}\right)^{y}\left(\frac{1}{6}\right)^{4-x-y}.\]

Key Concepts

Discrete Random Variables

When you hear about discrete random variables, think about a list of outcomes that can be counted. For example, rolling a die has six possible outcomes, which are discrete and finite in number. In probability theory, a discrete random variable is a type of variable that can take on a countable number of distinct values.

Each value of a discrete random variable is associated with a certain probability. Consider the case where you throw a fair six-sided die; the result of the throw is a discrete random variable. Each side of the die (1 through 6) has an equal probability of 1/6. The key property of discrete random variables lies in the fact that we can list all the possible outcomes along with their probabilities, which can be displayed in a table called a probability mass function.

Joint Probability Distribution

When dealing with two discrete random variables, like in our textbook example, you need to understand how one variable can affect the other. This relationship is expressed by the joint probability distribution. It's a mathematical tool that maps the likelihood of both events, or random variables, occurring simultaneously.

To put it in simpler terms, imagine having two dice instead of one: a red die and a blue die. The outcome of rolling each die is a random variable. The joint probability distribution of these two variables would tell us the probability of rolling any combination of numbers on the two dice, such as a 2 on the red die and a 5 on the blue die. This distribution is crucial for computing marginal probabilities, as it considers the outcomes of both variables together, and is often visualized through a table or a formula.

Probability Mass Function

The probability mass function (PMF) is the function that gives us the probability associated with each possible value of a discrete random variable. In essence, the PMF is a ‘who’s who’ list for probabilities, mapping each outcome of a discrete random variable to its probability.

A PMF must satisfy two conditions: firstly, all probabilities it assigns must be between 0 and 1, reflecting the fundamental law that probabilities can never be negative or more than certain; and secondly, the sum of all these probabilities must be equal to 1 – after all, the certainty that some outcome will occur is absolute. Thus, the PMF serves as a foundational concept, not only in our textbook problem but also in the entire study of probability and statistics.