Problem 250 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that $$ f_{Y \mid x}(y)=\frac{2 y+4 x}{1+4 x} \quad \text { and } \quad f_{X}(x)=\frac{1}{3} \cdot(1+4 x) $$ for \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\). Find the marginal pdf for \(Y\).

Step-by-Step Solution

Answer

The marginal pdf for \(Y\) is given by the integral from 0 to 1 of \( \frac{2y+4x}{3}\) dx. After calculation, this gives the function for the marginal pdf of \(Y\).

Step 1: Setting Up the Integral

The marginal pdf for \(Y\) is given by \(f_Y(y) = \int f_{Y \mid X}(y \mid x)f_X(x) dx\). Therefore, plug in the given functions into this formula, the integral to calculate becomes: \(\int_0^1 \frac{2y+4x}{1+4x} \cdot \frac{1}{3} \cdot (1+4x) dx\).

Step 2: Simplifying the Integrant

By multiplying the two functions inside the integral, the expression inside the integral becomes \( \frac{2y+4x}{3}\).

Step 3: Solving the Integral

When integrating from 0 to 1, the antiderivative of the integrant can be calculated by using standard antiderivative rules. After calculating the integral, you subtract the value at 0 from the value at 1.

Step 4: Calculation of the Marginal PDF

After evaluating the antiderivative at 1 and subtracting the value at 0, you obtain the marginal pdf of \(Y\).

Key Concepts

Understanding Conditional Probability

Conditional probability refers to the likelihood of an event occurring based on the occurrence of another event. This concept is prevalent in many statistical analyses where relationships between different events are examined. In the exercise provided, we have a conditional probability represented by the function \(f_{Y \mid x}(y)\). This function describes the probability distribution of \(Y\) given \(x\) has already occurred.

• If we want to find the probability of \(Y\) occurring given a certain \(x\), we use this function.
• This is crucial in scenarios where events influence each other and are not independent.

For example, suppose \(f_{Y \mid x}(y)\) evaluates a scenario where you need the probability of rain given that clouds are present. Here, the presence of clouds alters the probability of rain, hence why we consider it conditionally.

Analyzing Marginal Distribution

Marginal distribution is a way to understand the overall probability distribution of one variable within a joint distribution. That means, it answers the question of how values are distributed across a single variable, independent of other variables.

• For the variable \(Y\), we find the marginal distribution by integrating out the variable \(X\).
• Mathematically, the marginal distribution of \(Y\), denoted \(f_Y(y)\), is expressed as \(\int f_{Y \mid X}(y \mid x)f_X(x) dx\).

In the exercise, we determine \(f_Y(y)\), allowing us to see how \(Y\) behaves overall, not confined by specific conditions pertaining to \(x\). This reveals the independent behavior of \(Y\) across its entire range.

Exploring Probability Density Function (PDF)

A probability density function (PDF) provides a way to describe the likelihood of a continuous random variable. It tells us how the probabilities are distributed over the values.

• The PDF, denoted as \(f(x)\), gives the probability per unit on the x-axis.
• It does not give the probability directly, rather it is used to calculate it over intervals.

In our exercise, the functions \(f_{Y \mid x}(y)\) and \(f_X(x)\) are both PDFs. Analysis of these allows us to understand how each variable behaves and relates to others (conditional on \(X\) or not). Integrating a PDF over its range yields the total probability or aids in finding marginal distributions.

Mastering Integration in Probability

Integration in probability is a key mathematical tool that helps translate a PDF into interpretable probabilities. By integrating a PDF over a certain interval, we obtain the probability that the variable falls within that interval.

• The technique is critical when dealing with continuous random variables.
• In the context of the exercise, integration allows us to move from joint distributions to marginal distributions.

The process of integrating \(\int_0^1 \frac{2y+4x}{3} dx\) to find \(f_Y(y)\) exemplifies using integration to gather complete insights into the behavior of \(Y\). The integration acts to sum up all probabilities across the domain of \(X\), yielding an overall probability profile for \(Y\).