Problem 150 - An Introduction to Mathematical Statistics and Its Applications

Question

For each of the following joint pdfs, find \(f_{X}(x)\) and \(f_{Y}(y)\). (a) \(f_{X, Y}(x, y)=\frac{1}{2}, 0 \leq x \leq y \leq 2\) (b) \(f_{X, Y}(x, y)=\frac{1}{x}, 0 \leq y \leq x \leq 1\) (c) \(f_{X, Y}(x, y)=6 x, 0 \leq x \leq 1,0 \leq y \leq 1-x\)

Step-by-Step Solution

Answer

For problem (a) \(f_X(x) = \frac{2-x}{2}\) where \(0 \leq x \leq 2\) and \(f_Y(y) = \frac{y}{2}\) where \(0 \leq y \leq 2\). For problem (b) \(f_X(x) = x\) where \(0 < x \leq 1\) and \(f_Y(y) = 1\) where \(0 \leq y < 1\). For problem (c) \(f_X(x) = 6x - 6x^2\) where \(0 \leq x \leq 1\) and \(f_Y(y) = 3 - 6y\) where \(0 \leq y \leq 0.5\).

Step 1: Problem (a) Solution

Given, the joint pdf \(f_{X, Y}(x, y) = \frac{1}{2}\), where \(0 \leq x \leq y \leq 2\). To find \(f_X(x)\), integrate the joint pdf over the range of \(y\), from \(x\) to \(2\). \[f_X(x) = \int_x^2 f_{X, Y}(x, y) dy = \int_x^2 \frac{1}{2} dy = [\frac{1}{2}y]_x^2 = \frac{2-x}{2}\] Notice that \(f_X(x)\) is defined when \(0 \leq x \leq 2\). To get \(f_Y(y)\), integrate over the range of \(x\), from \(0\) to \(y\). \[f_Y(y) = \int_0^y f_{X, Y}(x, y) dx = \int_0^y \frac{1}{2} dx = [\frac{1}{2}x]_0^y = \frac{y}{2}\] Here, \(f_Y(y)\) is defined when \(0 \leq y \leq 2\).

Step 2: Problem (b) Solution

Given, the joint pdf \(f_{X, Y}(x, y) = \frac{1}{x}\), where \(0 \leq y \leq x \leq 1\). To find \(f_X(x)\), integrate over \(y\)'s range, from \(0\) to \(x\). \[f_X(x) = \int_0^x f_{X, Y}(x, y) dy = \int_0^x \frac{1}{x} dy = [y]_0^x = x\] This means \(f_X(x)\) is defined when \(0 < x \leq 1\). \(f_Y(y)\) is calculated differently as it has an infinite discontinuity. The outcome is \(f_Y(y) = 1\), where \(0 \leq y < 1\).

Step 3: Problem (c) Solution

Given, the joint pdf \(f_{X, Y}(x, y) = 6x\), where \(0 \leq x \leq 1\) and \(0 \leq y \leq 1-x\). To find \(f_X(x)\), integrate over \(y\)'s range, from \(0\) to \(1-x\). \[f_X(x) = \int_0^{1-x} f_{X, Y}(x, y) dy = \int_0^{1-x} 6x dy = [6xy]_0^{1-x} = 6x-6x^2\] This works when \(0 \leq x \leq 1\). For \(f_Y(y)\), \(f_Y(y) = \int_y^{1-y} f_{X, Y}(x, y) dx = 3-6y\). This is defined when \(0 \leq y \leq 0.5\).

Key Concepts

Joint PDF

A Joint Probability Density Function (Joint PDF) describes how two random variables interact with each other. It represents the likelihood of two variables, say \(X\) and \(Y\), taking on specific values simultaneously. For example, if we have a joint PDF \(f_{X, Y}(x, y)\), it tells us the probability density at the point \((x, y)\).
To work with joint PDFs, you often need to calculate other functions like marginal distributions, which involve integrating over one of the variables. To find insights about individual variables from a joint PDF, integration plays a critical role. Joint PDFs must satisfy two main properties:

• The total probability over the entire space must be 1.
• The joint PDF must be non-negative for all values of \(x\) and \(y\).

Understanding these properties helps in examining how probability mass is distributed over two random variables. Calculating the marginal distributions helps in getting detailed information about each variable independently from their joint behavior.

Marginal Distribution

To understand individual behaviors of random variables from a Joint PDF, Marginal Distributions come into play. A marginal distribution provides the probability distribution of a single variable by integrating out the other variable.
For instance, if we have a joint PDF \(f_{X, Y}(x, y)\) and want to find the marginal distribution for \(X\), we integrate over the possible values of \(Y\). Mathematically, this is described as:
\[f_X(x) = \int_{a}^{b} f_{X, Y}(x, y) \, dy\]
Where \(a\) and \(b\) are the limits within which \(Y\) can vary given \(x\). Similarly, for \(Y\), we integrate over \(X\):
\[f_Y(y) = \int_{c}^{d} f_{X, Y}(x, y) \, dx\]
Where \(c\) and \(d\) correspond to the range of \(X\) for given values of \(Y\). Hence, marginal distributions allow us to focus on individual behaviors while ignoring the influence of another variable.

Integration in Probability

Integration is a fundamental tool in probability theory for calculating probabilities and deriving meaningful statistics from continuous random variables. It is especially critical when dealing with Probability Density Functions (PDFs) and Joint PDFs.
One primary use of integration is finding Marginal Distributions from Joint PDFs, allowing us to focus on one random variable at a time. For instance, integrating a joint PDF over a variable's entire range gives us insights into another variable's behavior.
Moreover, integration helps in ensuring PDFs are properly normalized. This requires integrating the PDF over its entire defined space and equating it to 1:
\[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_{X, Y}(x, y) \, dx \, dy = 1\]
This guarantees that the total probability over the space sums to unity, fulfilling a basic property of Probability Theory. Therefore, proper application of integration methods is crucial in achieving mathematically sound probabilistic interpretations.

Probability Theory

Probability Theory forms the backbone of understanding how likely events are to occur. It provides a mathematical framework to study random events and outcomes. Concepts like Joint PDFs, Marginal Distributions, and the utilization of integration all draw from this fundamental theory.
Some essential principles include:

• Random Variables: Variables that take on values based on the outcomes of a random event.
• Probability Distribution: A function that maps outcomes of a random variable to their likelihood.
• Expected Value and Variance: Metrics used to summarize characteristics of probability distributions.

Probability Theory helps in creating models that describe real-world phenomena where uncertainty is inherent. Joint PDFs offer insights when dealing with multiple variables, while Marginal Distributions allow us to focus specifically on individual behaviors. Mastering these concepts enable students to tackle problems involving randomness with confidence.