I am trying to reason about the following scenario:

Let us have three random variables: $X$, $Y$, $Z$. $Y$ is independent of $Z$. Let us also have the following CDF's:

$$F_X, F_{X \mid Y}, F_{X \mid Z}$$

With the relationships:

$$F_X \neq F_{X \mid Y}\neq F_{X \mid Z}$$

I am trying to reason about what we can expect $F_{X \mid Y, Z}$ to be, given the above information. My intuition (and this was suggested in a comment here) is that $F_{X \mid Y, Z}$ can be expected to a be a weighted average of $F_{X \mid Y}$ and $F_{X \mid Z}$:

$$F_{X \mid Y, Z} = 0.5F_{X \mid Y} + 0.5F_{X \mid Z}$$

The reasoning for this is that if both $Y$ and $Z$ affect $X$, then knowing that both $Y$ and $Z$ occurred it seems reasonable to expect the total affect on $X$ to be partly due to $Y$ and partly due $Z$. Even though this seems reasonable, I have not been able to find a source/any derivation that this is indeed correct reasoning (hence this question).


An example may make this more concrete. Consider the scenario where we are trying to model shipping prices. Let $X$ be price, $Y$ be weight and $Z$ be state tax. We are given that weight and state tax are independent of each other (i.e. the weight of a shipment does not affect the state tax that it is being shipped to). We also are given that weight affects price (heavier objects cost more to ship), and state tax affects price (different states cost more or less to ship to). Let us now be given that the object we are trying to ship has a high weight and a state with low tax. Our CDF's $F_{price \mid \text{high weight}}$ and $F_{price \mid \text{low state tax}}$ then look like:

enter image description here

Where we can see clearly (as you would expect) that $F_{price \mid \text{high weight}}$ is shifted to the right compared to $F_{price \mid \text{low state tax}}$ (i.e. it has a higher average price).

Now let us consider how $price$ responds when we know both we have a low state tax and a high weight. Intuitively it seems as though we would expect our price to be increased due to the high weight, and decreased due to the low tax, meaning our price would be a weighted average in essence.

This would like like:

$$F_{price \mid \text{high weight, low state tax}} = 0.5F_{price \mid \text{low state tax}} + 0.5F_{price \mid \text{high weight}}$$

Where visually we can see $F_{price \mid \text{high weight, low state tax}}$ is a weighted average of $F_{price \mid \text{low state tax}}$ and $F_{price \mid \text{high weight}}$:

enter image description here

Subsequently, if we observe that $F_{price \mid \text{high weight, low state tax}}$ is actually far different than $0.5F_{price \mid \text{low state tax}} + 0.5F_{price \mid \text{high weight}}$, we can conclude that their is some interaction occurring (for instance maybe shipments of a high weight get an additional tax when shipped to this particular state, hence shifting our CDF $F_{price \mid \text{high weight, low state tax}}$).

Additional Resources

For anyone interested, I have read through the following related posts, but none have quite answered the question I am posting here.

  • 1
    $\begingroup$ You are mistaken in your plot of a weighted average: you have averaged the values along the horizontal coordinate, but the intended meaning is an average along the vertical coordinate. The weighted average of your two extreme CDFs would be one that gradually slopes from the bottom left to the upper right. $\endgroup$
    – whuber
    Commented Sep 22, 2020 at 16:48
  • $\begingroup$ @whuber Thanks for catching that/pointing out my error! I edited the plot so it should be correct now. With that said, are you able to comment on the main question of this post: Is it reasonable to expect that $F_{X \mid Y, Z} = 0.5F_{X \mid Y} + 0.5F_{X \mid Z}$? $\endgroup$
    – ndake11
    Commented Sep 22, 2020 at 19:11
  • $\begingroup$ It might be reasonable when additional assumptions are made. As a general proposition this doesn't look even like a good approximation: why can't $Y$ and $Z$ interact to make $F_{X\mid Y,Z}$ entirely unlike either of the marginal conditional distributions? $\endgroup$
    – whuber
    Commented Sep 22, 2020 at 21:06
  • $\begingroup$ @whuber $Y$ and $Z$ can interact, I am just trying to determine if it reasonable to state that in the case where $Y$ and $Z$ don't interact the distribution of $F_{X \mid Y, Z}$ would approximately be $0.5 F_{X \mid Y} + 0.5 F_{X \mid Z}$. Given no interaction between $Y$ and $Z$, does that seem like a reasonable approximation to you? If not, what additional assumptions would you require for it to be? Or do you have another approximation in mind altogether? $\endgroup$
    – ndake11
    Commented Sep 23, 2020 at 21:39
  • 1
    $\begingroup$ You can transform your data into $p(Y|X)$ and $p(Z|X)$, and then use copulas to guess $p(Y,Z|X)$. But the example seems artificial -- what are the real $X$, $Y$, and $Z$ that you care about, and what single number that you would compute from $F(X|Y,Z)$ would be of greatest interest? $\endgroup$
    – user225256
    Commented Oct 4, 2020 at 10:54


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.