The marginal probability density function of X  denoted by

The marginal probability density function (PDF) of a random variable \( X \), denoted by \( f_X(x) \), represents the probability density of \( X \) while integrating out other random variables if \( X \) is part of a joint distribution. Essentially, it provides the likelihood of \( X \) taking on a specific value \( x \), regardless of the values of other variables in the system.

If \( X \) and \( Y \) are jointly distributed random variables with joint PDF \( f_{X,Y}(x,y) \), the marginal PDF of \( X \) is obtained by integrating out \( Y \) from the joint PDF:

\[
f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x,y) \, dy
\]

This gives the probability density for \( X \) at a specific point \( x \), considering all possible values of \( Y \).

Would you like further details on how marginal PDFs are used, or how they relate to conditional distributions?

Here are 5 related questions that might interest you:

1. How is the marginal PDF related to the joint PDF?
2. Can you explain the difference between marginal and conditional PDFs?
3. What are the properties of a marginal probability density function?
4. How do you compute the marginal PDF in multivariable distributions?
5. How does the concept of independence affect the marginal PDF?

**Tip:** Always remember that the total area under the marginal PDF of \( X \) must equal 1, ensuring it represents a valid probability distribution.

The marginal probability density function of X denoted by

Understand how to compute the marginal probability density function (PDF) of a random variable X from its joint distribution with another variable Y. This detailed explanation includes key concepts in probability theory, the integral formula, and its relationship to the joint PDF. Suitable for undergraduate students studying probability and statistics.

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