Consider the pair of continuous random variables defined on the same sample space. What is the conditional distribution in terms of the joint distribution function and the marginal distributions? (Select all that apply)

A. The joint PDF divided by the product of the two marginal PDFs
B. The joint PDF divided by the sum of the two marginal PDFs
C. The joint PDF divided by the marginal PDF of Y
D. The joint PDF divided by the marginal PDF of X

In the context of computers and technology, understanding conditional distributions in terms of joint and marginal distributions can be applied to various fields such as machine learning, data analysis, and statistical modeling. Here's how the conditional distribution relates to joint and marginal distributions in this context:

1. Joint PDF (Probability Density Function): Represents the probability distribution of two random variables occurring together. In machine learning, this might be the distribution of features and target labels in a dataset.

2. Marginal PDF: Represents the probability distribution of a single variable irrespective of the other. In data analysis, this might be the distribution of just the features or just the target labels.

3. Conditional PDF: Represents the probability distribution of one variable given that another variable is known. In predictive modeling, this could be the distribution of the target variable given the features.

Given two continuous random variables X and Y:

- The joint PDF is [tex]$f_{X,Y}(x,y)$[/tex].

- The marginal PDFs are [tex]$f_X(x)$ for $X$ and $f_Y(y)$ for $Y$.[/tex]

The conditional distribution can be defined as:

1. Conditional PDF of Y given X = x:

This tells us how Y is distributed if we know the value of X.

2. Conditional PDF of X given Y = y:

[tex]\[f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}\][/tex]

This tells us how X is distributed if we know the value of Y.

In terms of the given options:

- Option A (The joint PDF divided by the product of the two marginal PDFs) is incorrect. This would not give a conditional distribution.

- Option B (The joint PDF divided by the sum of the two marginal PDFs) is incorrect. This is not a standard formula in probability theory.

- Option C (The joint PDF divided by the marginal PDF of Y is correct for finding the conditional PDF of X given Y = y.

- Option D (The joint PDF divided by the marginal PDF of X is correct for finding the conditional PDF of Y given X = x.

Therefore, in terms of computer and technology, for calculating conditional distributions using joint and marginal PDFs,

Therefore, the correct answers are:

C. The joint PDF is divided by the marginal PDF of Y.

D. The joint PDF is divided by the marginal PDF of X.