Suppose there are two random variables $X$ and $Y$, and I was given the domain of the marginal distribution $f_X(x)$ and conditional distribution $f_{XY}(y|x)$ (which is a function of x). How do I find the domain of the joint distribution $f(x,y)$?

What I tried so far is to take the domain of the marginal distribution of $x$ as the domain of $x$ in the joint distribution. For $y$, I just plugged in values of $x$ and the resulting interval is the domain of $y$. Is this correct? If not, what's the right approach? Thank you

  • $\begingroup$ You are correct in the sense that $f(x,y)$ will always be zero outside of the rectangle that you've constructed (Cartesian product of interval in $x$ and interval in $y$). But $f(x,y)$ may also be zero in parts of the inside of the rectangle. $\endgroup$
    – angryavian
    Commented Jan 12, 2022 at 3:22

1 Answer 1


For a joint probability distribution the variables have to share the same probability space (ie they have to have a set of all possible outcomes, they have a set of events(measurements), they have to have a probability/ probability function for all events). That means 2 balls with each ball being either red or green, the width and length of a box. Say for a single ball red or green we have a single variable and no need for joint.

When one variable is continuous and one is discrete we have mixed probability

๐‘“๐‘ฆ|๐‘ฅ(๐‘ฆ|๐‘ฅ)=fxy * fx Or ๐‘“x|y(x|y)=fxy * fy

This means that all probabilities of y have a x probability as well(even if it is zero). But just becz Xmax is max and xmin is min doesnโ€™t mean the event that links them is a ymax ymin ie (xmax, notymax) and (xmin, notymin) [but fx fxy and fy are always between 0 and 1 becz they are probabilities ie a domain is not a range]

So for flower petals we have length a continuous variable and color red or blue a discrete. To get the full interval of lengths x we need all data points of red and of blue


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