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.