The question asks about the relationship between $F_X(x)$ and $F_Y(y)$, where $F$ is a cumulative distribution function (CDF), and it specifies that $x \leq y$.

In general, the properties of the CDF, $F$, are:

1. $F_X(x) = P(X \leq x)$, meaning it represents the probability that the random variable $X$ takes a value less than or equal to $x$.

2. A CDF is a non-decreasing function. Therefore, if $x \leq y$, then $F(x) \leq F(y)$.

Thus, for $x \leq y$, the relationship is: $F_X(x) \leq F_Y(y)$ if $X$ and $Y$ share the same distribution. If they do not, additional conditions would need to be considered.

Would you like further clarification or examples? Here are some related questions to expand on this topic:

1. What is the significance of the cumulative distribution function in probability?
2. How is the relationship $F_X(x) \leq F_Y(y)$ derived from the properties of the CDF?
3. Can $F_X(x)$ ever decrease as $x$ increases?
4. What happens if $X$ and $Y$ are independent? How would $F_X(x)$ and $F_Y(y)$ interact?
5. How does this principle apply to joint cumulative distribution functions?

Tip: Remember that the CDF always lies between 0 and 1 and is right-continuous!