How to derive conditional distribution function on range from conditional distribution function on a single value

Question

Suppose we know $$f_{X|Y}(x|y)$$ which is the conditional distribution function of $$X=x | Y=y$$ how to derive the conditional probability function of $$X=x|y_1 < Y < y_2$$? Should it be $$\int_{y_1}^{y_2} f_{X|Y}(x|y) dy$$? I am new to statistics and your input is greatly appreciated, thanks!

Answer 1

Suppose the joint pdf of $(X, Y)$ is given by $f(x, y)$, and $P(y_1 < Y < y_2) > 0$, I am assuming you are interested in determining the conditional distribution of $X$ given the event $B: = \{y_1 < Y < y_2\}$ of positive probability, i.e., $P(X \leq x | y_1 < Y < y_2)$, which by the conditional probability definition (with respect to events) $P(A|B) = \frac{P(A \cap B)}{P(B)}$ is \begin{align*} P(X \leq x | y_1 < Y < y_2) = \frac{P(X \leq x, y_1 < Y < y_2)}{P(y_1 < Y < y_2)} = \frac{\int_{-\infty}^x\int_{y_1}^{y_2}f(s, y)dyds}{\int_{y_1}^{y_2}f_Y(y)dy}. \tag{1}\label{1} \end{align*} If you want the conditional density given $B$, differentiating $\eqref{1}$ with respect to $x$ yields \begin{align*} f(x | y_1 < Y < y_2) = \frac{\int_{y_1}^{y_2}f(x, y)dy}{\int_{y_1}^{y_2}f_Y(y)dy} = \int_{y_1}^{y_2}\frac{f(x, y)}{C}dy, \tag{2}\label{2} \end{align*} where $C = \int_{y_1}^{y_2}f_Y(y)dy = P(B)$. $\eqref{2}$ also shows that $f(x | y_1 < Y < y_2)$ does not link to $f_{X|Y = y}(x, y)$ in a simple way (as your title queries) unless you rewrites the numerator $f(x, y)$ of the integrand in $\eqref{2}$ as $f_Y(y)f_{X|Y = y}(x|y)$, but I consider this rearrangement unnecessary and less transparent, with the viewpoint that the joint distribution $f(x, y)$ is the most fundamental law to describe the interrelation of $(X, Y)$ while $f_{X|Y = y}(x|y)$ is just one of its derivatives.

Note how $\eqref{2}$ differs from what you proposed, which is $g(x) := \int_{y_1}^{y_2}\frac{f(x, y)}{f_Y(y)}dy$. It is easy to verify that while $\eqref{2}$ is a valid density function (i.e., it integrates to $1$), $g(x)$ is not.

You may glean more insight on the concept of conditioning on an event with positive probability in this post. In my opinion, this concept is generally easier than the concept of conditioning on a single-value, which requires a good understanding of measure-theoretic probability to fully grasp.