Suppose that 𝑋 and 𝑌 are continuous with known PDFs 𝑓𝑋 and 𝑓𝑌. Find the values of 𝐴 and 𝐵 to solve for 𝑓𝑍|𝑌(𝑧|𝑦) in terms of 𝑓𝑋.
To find the conditional probability density function (PDF) 𝑓𝑍|𝑌(𝑧|𝑦) in terms of 𝑓𝑋, we need to use the concept of conditional probability and the joint probability density function (PDF). Let's go step by step:
1. Start with the definition of conditional probability: 𝑓𝑍|𝑌(𝑧|𝑦) = 𝑃(𝑍 = 𝑧|𝑌 = 𝑦).
2. Use Bayes' theorem to write the conditional probability in terms of joint probabilities: 𝑃(𝑍 = 𝑧|𝑌 = 𝑦) = 𝑃(𝑍 = 𝑧, 𝑌 = 𝑦) / 𝑃(𝑌 = 𝑦).
3. Express the joint probability in terms of marginal PDFs: 𝑃(𝑍 = 𝑧, 𝑌 = 𝑦) = 𝑓𝑍,𝑌(𝑧, 𝑦).
4. Similarly, write the marginal probability as 𝑃(𝑌 = 𝑦) = 𝑓𝑌(𝑦).
5. Finally, substitute the expressions for joint and marginal probabilities into the conditional probability formula: 𝑓𝑍|𝑌(𝑧|𝑦) = 𝑓𝑍,𝑌(𝑧, 𝑦) / 𝑓𝑌(𝑦).
At this point, we have expressed the conditional PDF 𝑓𝑍|𝑌(𝑧|𝑦) in terms of the joint PDF 𝑓𝑍,𝑌(𝑧, 𝑦) and the marginal PDF 𝑓𝑌(𝑦).
Note that in order to proceed further and explicitly solve for 𝑓𝑍|𝑌(𝑧|𝑦) in terms of 𝑓𝑋, we need additional information about the relationship between 𝑋, 𝑌, and 𝑍. Without such information, we cannot fully determine 𝑓𝑍|𝑌(𝑧|𝑦) solely based on 𝑓𝑋.
To find the conditional PDF 𝑓𝑍|𝑌(𝑧|𝑦), we can use the conditional probability formula:
𝑓𝑍|𝑌(𝑧|𝑦) = 𝑓𝑍,𝑌(𝑧,𝑦) / 𝑓𝑌(𝑦)
However, we need 𝑓𝑍,𝑌(𝑧,𝑦) for this formula. To find 𝑓𝑍,𝑌(𝑧,𝑦), we need the joint PDF of 𝑋 and 𝑌, denoted as 𝑓𝑋,𝑌(𝑥,𝑦).
Assuming 𝑋 and 𝑌 are independent, we can express their joint PDF as the product of their individual PDFs:
𝑓𝑋,𝑌(𝑥,𝑦) = 𝑓𝑋(𝑥) * 𝑓𝑌(𝑦)
Now, substituting this into the conditional probability formula, we have:
𝑓𝑍|𝑌(𝑧|𝑦) = 𝑓𝑋,𝑌(𝑧,𝑦) / 𝑓𝑌(𝑦)
= (𝑓𝑋(𝑧) * 𝑓𝑌(𝑦)) / 𝑓𝑌(𝑦)
Finally, simplifying this expression, we get:
𝑓𝑍|𝑌(𝑧|𝑦) = 𝑓𝑋(𝑧)
Therefore, 𝑓𝑍|𝑌(𝑧|𝑦) is equal to 𝑓𝑋(𝑧), regardless of the values of 𝐴 and 𝐵.