Problem 242 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose \(X, Y\), and \(Z\) have a trivariate distribution described by the joint pdf $$ p_{X, Y, Z}(x, y, z)=\frac{x y+x z+y z}{54} $$ where \(x, y\), and \(z\) can be 1 or 2 . Tabulate the joint conditional pdf of \(X\) and \(Y\) given each of the two values of \(z\).

Step-by-Step Solution

Answer

The exercise involves finding conditional pdfs. First, calculate the marginal pdfs of \(Z\), then use these values to calculate the conditional pdfs of \(X\) and \(Y\) given \(Z=1\) and \(Z=2\). Tabulate the results for all combinations.

Step 1: Calculate the Probability of Z for both z=1 and z=2

The marginal pdf of \(Z=z\) can be calculated using \(p_Z(z) = \sum_{x=1}^{2} \sum_{y=1}^{2} p_{X,Y,Z}(x,y,z)\). So, calculate \(p_Z(1)\) and \(p_Z(2)\).

Step 2: Calculate the Conditional Probability for X and Y Given Z=z

Now calculate the conditional pdf of \(X\) and \(Y\) given \(Z=z\). Use the formula \(p_{X,Y | Z}(x,y | z) = \frac{p_{X,Y,Z}(x,y,z)}{p_Z(z)}\). Compute this for all combinations of \(X, Y\) given \(Z=1\) and \(Z=2\).

Step 3: Tabulate the Conditional PDF of X and Y given Z=z

Finally, tabulate the calculated conditional pdf values. Create a table with rows for the values of \(X\) and \(Y\) and columns for the two values of \(Z\).

Key Concepts

Probability Density Function

A probability density function (PDF) is a fundamental concept in probability and statistics. It provides a way to describe the distribution of continuous random variables. For a given random variable, the PDF represents a function that describes the likelihood of that variable taking on a certain range of values.
In the context of a trivariate distribution involving variables like \(X\), \(Y\), and \(Z\), the joint PDF, \(p_{X,Y,Z}(x,y,z)\), is used to indicate the probability density across the three variables. This specific trivariate PDF is given by \(\frac{x y + x z + y z}{54}\). This function essentially combines the contributions of each variable's interactions.
- The function is normalized, meaning the integral over all space should equal 1. Therefore, ensuring valid probability measures.- It is crucial in calculating other forms of probabilities like conditional and marginal probabilities.
In simpler terms, think of the PDF as a map that shows the "density" of where values are likely to occur in the space defined by \(X\), \(Y\), and \(Z\). This helps in understanding how these variables interact with each other probabilistically.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has occurred. When working with joint distributions like with variables \(X\), \(Y\), and \(Z\), understanding conditional probabilities becomes essential.
To find the conditional probability, we need to determine how the probability distribution changes when we know a certain event has occurred. For example, calculating the conditional PDF of \(X\) and \(Y\) given \(Z=z\) involves:

• First, determine the marginal probability of \(Z=z\), using all combinations of \(X\) and \(Y\).
• Then, use the formula \(p_{X,Y | Z}(x,y | z) = \frac{p_{X,Y,Z}(x,y,z)}{p_Z(z)}\) to find the conditional PDF for different values of \(z\).

The above formula essentially tells us how \(X\) and \(Y\)'s distribution adjusts in light of knowing the particular value \(Z\) takes.
This application of conditional probability is vital in various fields like statistics, machine learning, and actuarial sciences, where understanding dependencies between variables is crucial.

Marginal PDF

The marginal probability density function is a way of expressing the probability distribution of a subset of collection variables from a joint distribution. For instance, when dealing with multiple variables like \(X\), \(Y\), and \(Z\), you might want to focus on one variable only, say \(Z\), without considering specific values of \(X\) and \(Y\). This is where the marginal PDF comes in.
To calculate it, you sum over the other dimensions (variables) in the joint distribution. Specifically, for the marginal PDF of \(Z\), you would sum: \[ p_Z(z) = \sum_{x=1}^{2} \sum_{y=1}^{2} p_{X,Y,Z}(x,y,z) \] This results in a function that conveys the probability distribution of the variable \(Z\) by itself.
Understanding the marginal PDF helps simplify complex probabilistic models by focusing on individual variables' distributions. It's also essential for calculating conditional probabilities, as it serves as the denominator in the formula for conditional PDFs. - Marginal PDFs are extensively used in multivariate statistics to unpack things into more manageable bits.- They are crucial in real-world applications where one might be only interested in the behavior of a single variable while considering the presence of others. By reducing the scope of analysis to just one variable, you gain clearer insights into its standalone behavior, disconnected from other variables.