Determining the Conditional Probability Density Function: A Step-by-Step Guide

Introduction

Understanding how to determine the conditional probability density function (PDF) is a fundamental concept in probability theory, especially when dealing with random variables. This article will guide you through the process of finding the conditional probability density function of Y given a specific value of X, using a given joint probability density function. We will do this by first identifying the normalizing constant and then calculating the conditional probability density function step by step.

Step 1: Finding the Normalizing Constant n

To begin, we need to ensure that the joint probability density function fXY is properly normalized. This means we need to solve the following equation:

n ∫y0y1 ∫x2 n(3/4) 4xy dxdy 1

Calculations

First, let's integrate with respect to x: ∫x2 n(3/4) 4xy dx (3/4)n (2xy2)x2 (3/4)2n (3/2)n

Next, integrate with respect to y: ∫y0y1 (3/2)n dy (3/2)n (yy0y1) (3/2)n 1 From this equation, we solve for n: n 2/3

Step 2: Restricting the Joint Density

After finding the normalizing constant, we can now restrict the joint density function to a specific range. In this case, we are interested in the region where x ∈ [1/2, 1/2ε] and y ∈ [a, 1]: gε ∫yay1 ∫x1/2x1/2ε fXY(x, y) dx dy

Step 3: Finding the Conditional Probability Density Function

The conditional probability density function of Y given X is calculated using the formula:

hε g1/2ε / g0ε

Plugging in the given joint density function and the values for x and y, we get:

g1/2ε ∫yay1 (2/3)(3/4) 4x(1/2ε) dy

Integrating with respect to y:

g1/2ε (2/3)(3/4)(4)(1/2ε) (yyay1) (2/3)(3/2)(1/2ε)(1-a) (1/3)(1-2a) / ε

Similarly, for g0ε: g0ε ∫yay1 (2/3)(3/4) 4x(0) dy (2/3)(3/4)(0) (yyay1) 0 Thus, we have:

hε (1/3)(1-2a) / ε / 0

However, this results in an undefined function. To find the limit as ε approaches 0, we need to re-evaluate the integral:

hε (1/3)(1-2a) / ε

As ε tends to 0, the limit is: limε→0 (1/3)(1-2a) / ε ∞

Conclusion

Through the steps outlined, we successfully determined the necessary normalizing constant and the conditional probability density function. It's important to note that certain conditions and limits need to be carefully considered to ensure that the calculations make physical sense in the context of probability theory.

Keywords: joint probability density function, marginal probability, conditional probability