Problem 165 - An Introduction to Mathematical Statistics and Its Applications

Question

Calculate \(p_{X, Y}(0,1)\) if \(p_{X, Y, Z}(x, y, z)=\) \(\frac{3 !}{x[y ! z !(3-x-y-z) !}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{12}\right)^{y}\left(\frac{1}{6}\right)^{z} \cdot\left(\frac{1}{4}\right)^{3-x-y-z}\) for \(x, y, z=0,1\), 2,3 and \(0 \leq x+y+z \leq 3\).

Step-by-Step Solution

Answer

The calculation of \(p_{X, Y}(0,1)\) includes two parts where the Z variable takes two possible values (2 and 3). The short answer will be the sum of these two parts, which is the final result of \(p_{X, Y}(0,1)\).

Step 1: Identify Possible Values of Z

Since \(x+y+z<=3\), \(z<=3-x-y\). Since X = 0 and Y = 1 are given, Z can be 2 or 3.

Step 2: Calculate the PMF for each Value of Z

Substitute \(x=0, y=1\) and \(z=2\) and then \(z=3\) into \(p_{X, Y, Z}(x, y, z)\). Calculate the PMF for these two sets of values.

Step 3: Sum the PMFs

Add the two PMFs calculated in Step 2 to obtain the value of \(p_{X, Y}(0,1)\). This is because in a PMF, for a certain combination of variable values, the probabilities of all possible outcomes sum up to 1.

Key Concepts

Probability Mass Function (PMF)

In the world of probability and statistics, the Probability Mass Function (PMF) is a critical concept when dealing with discrete random variables. It provides the probability that a discrete random variable is exactly equal to some value. Simply put, if you want to know the likelihood of your variable taking on a specific value, the PMF is your go-to tool.
A PMF is expressed generally as a function that maps each possible outcome to a probability. These probabilities are non-negative and must sum to 1 over all possible outcomes, ensuring that one of the possible outcomes must occur. A PMF is particularly useful when you track each specific possible outcome of an experiment, like the number of heads in a series of coin flips.
In practical terms, consider the formula given for the PMF in the exercise: \[ p_{X, Y, Z}(x, y, z) = \frac{3!}{x[y! z!(3-x-y-z)!}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{12}\right)^{y}\left(\frac{1}{6}\right)^{z}\cdot\left(\frac{1}{4}\right)^{3-x-y-z} \]This complex expression evaluates the probability for variables X, Y, and Z in a joint distribution scenario, reflecting specific probability weights based on assigned values.

Random Variables

Random variables are a foundational aspect of probability and statistics, serving as numerical representations of the outcomes of random phenomena. These are called "random" because they originate from chance-related experiments. Imagine rolling a dice: the number you get is a random variable, as it can be any number from 1 to 6, depending on the roll.
Random variables can be classified into two types: continuous and discrete. In our context, we're concerned with discrete random variables, which take on distinct, separate values. They are often countable, such as the number of calls received in a day or points scored in a game.
In the exercise, the problem involves random variables X, Y, and Z, each representing some aspect of an experiment. These variables are part of a joint probability distribution, which refers to the probability distribution over more than one random variable. Understanding these variables' behavior is crucial for calculating joint probabilities and using the PMF.

Discrete Probability

Discrete probability focuses on scenarios where outcomes can be counted or listed outright. Calculating discrete probabilities involves scenarios like dice rolls, card draws, or other activities with a limited set of possible results.
One crucial feature of discrete probability is that it considers events with a finite number of outcomes. This is exemplified by our exercise's condition that \(0 \le x + y + z \le 3\). This boundary restricts the combinations of random variable outcomes we need to consider and simplifies calculations.
Another important piece is summation—adding up probabilities. In the provided step-by-step solution, we summed the probabilities for each potential value of Z to find \(p_{X, Y}(0,1)\). This is a key process in discrete probability, ensuring that all potential ways of achieving an event are accounted for, covering the complete spectrum of specific numeric outcomes. This ensures total probability equals 1, maintaining mathematical coherence and integrity.