ML Interview Q Series: Joint Density of Sum and Ratio of Uniform Variables via Jacobian Transformation
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11E-20. The random variables X and Y are independent and uniformly distributed on (0,1). Let V = X + Y and W = X/Y. What is the joint density of (V, W)? Are V and W independent?
Short Compact solution
Using the transformation (V, W) = (X+Y, X/Y) with inverse functions x = (v·w)/(1+w) and y = v/(1+w), we compute the absolute value of the Jacobian determinant and obtain v/(1+w)². Hence the joint density is
over the region 0 < w < ∞ and 0 < v < min(1 + w, 1 + 1/w), and 0 otherwise.
They are not independent.
Comprehensive Explanation
Derivation of the Joint Density
We start with two independent random variables X and Y, each uniform on (0,1). We define new variables:
V = X + Y
W = X / Y
Our goal is to find the joint density f_{V,W}(v, w). We do this by:
Writing down the inverse transformation that maps (v, w) back to (x, y).
Calculating the Jacobian of that inverse transformation.
Determining the region in the (v, w) plane for which (x, y) falls inside (0,1) × (0,1).
1) Inverse Transformation
From V = X + Y and W = X / Y, we solve for X and Y in terms of V and W:
Since W = X / Y, we have X = W·Y.
Since V = X + Y, we then have V = W·Y + Y = Y (W + 1).
Hence Y = V / (W + 1) and X = (V·W) / (W + 1).
So the inverse transformation is x = v·w / (1 + w), y = v / (1 + w).
2) Jacobian Determinant
We calculate the determinant of the Jacobian matrix for the mapping (v, w) → (x, y). When you perform this computation, you get
Jacobian = ∂(x, y) / ∂(v, w).
The determinant (before taking the absolute value) turns out to be -v / (1 + w)². Thus, its absolute value is
| -v / (1 + w)² | = v / (1 + w)².
3) Determining the Valid (v, w) Region
Because X and Y must lie in (0,1), the corresponding (v, w) values must satisfy:
0 < x = (v·w) / (1 + w) < 1,
0 < y = v / (1 + w) < 1,
and obviously V = x + y > 0, W = x / y > 0.
From x < 1 we get v·w / (1 + w) < 1, which rearranges to v < (1 + w) / w = 1 + 1/w.
From y < 1 we get v / (1 + w) < 1, which rearranges to v < 1 + w.
Since v = x + y must also be less than 2 (because x, y each < 1 individually), we collect these conditions into the single requirement:
0 < v < min(1 + w, 1 + 1/w), for w > 0.
Hence, the joint density is
Checking Independence
Two random variables V and W are independent if and only if their joint density factors into the product of their respective marginals over the entire valid domain. A quick inspection reveals that there is a complicated coupling between v and w through the region constraint 0 < v < min(1 + w, 1 + 1/w). In fact, it is straightforward to show that the joint density cannot be written as a product of a function of v alone and a function of w alone. Indeed, for uniform(0,1) X and Y, the sum and ratio do not simplify into independent pieces.
Thus, V and W are not independent.
Potential Follow-Up Questions
1) How would you integrate out W to find the marginal of V explicitly?
You would compute
f_V(v) = ∫ f_{V,W}(v, w) dw,
where w runs over the values that satisfy the constraints 0 < w < ∞ and v < min(1 + w, 1 + 1/w). This effectively splits into cases: one range for w < 1 and another for w > 1, because the function min(1 + w, 1 + 1/w) changes its form depending on whether w < 1, w = 1, or w > 1. You would then do piecewise integration to show that f_V(v) matches the well-known triangular shape for the sum of two Uniform(0,1) random variables:
f_V(v) = v for 0 < v < 1,
f_V(v) = 2 - v for 1 < v < 2,
and 0 otherwise.
2) Why intuitively are V = X + Y and W = X / Y not independent for uniform(0,1)?
Intuitively, knowledge of the sum of two numbers in (0,1) imposes some constraints on how large one of them can be relative to the other. For example, if you know the sum is close to 2, that forces each of X and Y to be close to 1, thereby restricting the possible range for the ratio X/Y to values near 1. Conversely, if the sum is very small, that forces the ratio to be more extreme if one variable is significantly larger than the other in that small range. Such constraints imply that the joint behavior of V and W cannot factor into separate, purely independent distributions.
3) Could you provide a small Python code snippet to confirm the non-independence numerically?
Below is a quick simulation in Python. We generate many samples of X, Y ~ Uniform(0,1), compute V and W, then check whether the sample correlation (for instance) between V and W is close to zero or not.
import numpy as np
N = 10_000_000
X = np.random.rand(N)
Y = np.random.rand(N)
V = X + Y
W = X / Y
corr = np.corrcoef(V, W)[0, 1]
print("Estimated correlation between V and W:", corr)
You will typically observe a correlation significantly away from zero, indicating dependence. (Correlation is just one simple measure; formal independence tests or PDFs confirm more rigorously that they are not independent.)
4) In contrast, for which distributions of X and Y might sum and ratio be independent?
A well-known example is when X and Y are i.i.d. Exponential(lambda). In that case, it can be shown that X/(X+Y) is independent of X+Y. The ratio X/(X+Y) is actually Beta(1,1) distributed (i.e., Uniform(0,1)), and X+Y has a Gamma(2, lambda) distribution, and they turn out to be independent. But for X and Y uniform(0,1), such independence does not hold.
5) Are there any pitfalls in applying the transformation approach for finding joint densities?
One of the most common pitfalls is forgetting to restrict to the valid region where the original (X, Y) remain in their support. Another common error is failing to include the absolute value of the Jacobian. Lastly, it is easy to overlook piecewise constraints that make the integration limits non-trivial. Always confirm the resulting distribution integrates to 1 and matches known properties for partial checks of correctness.
Below are additional follow-up questions
Could you discuss how the ratio W = X/Y might behave in edge cases when Y is very small, and how that impacts practical simulations or numerical stability?
When Y is near zero, the ratio X/Y can become arbitrarily large, since X/Y grows without bound as Y → 0 for a fixed X > 0. In a simulation setting, such extremely large values may cause numerical instabilities. For example, if Y is generated as a floating-point number extremely close to zero, the ratio might overflow to some representation of infinity.
In practice, you must decide how to handle such extreme outcomes. If your simulation language has built-in checks, you might see floating-point warnings or your program might produce NaNs. A common approach is to filter or clip values that exceed some threshold, but you must do that carefully so as not to bias the resulting estimates. Another approach is to reformulate the problem to avoid direct division by Y, perhaps by using a different transformation (like X / (X + Y)) if that suits the application and doesn’t introduce different numerical problems.
On a theoretical level, these edge cases are fully valid because Y is strictly in (0,1), so it is never exactly zero. However, from a practical standpoint, extremely small Y values can approximate zero well enough to cause floating-point issues. Always check whether your application can tolerate very large ratio outcomes or if you need a stable alternative representation.
How does the Jacobian-based method compare to other methods (e.g., the cumulative distribution function method) for deriving joint distributions under transformations?
The Jacobian-based “change of variables” formula is often the most direct way to find joint densities after a transformation. It systematically provides the correct factor f_{X,Y}(x, y) multiplied by the absolute determinant of the Jacobian. Its principal advantage is that it works in a straightforward, formulaic manner for continuous transformations that are one-to-one over the relevant domain.
The cumulative distribution function (CDF) method can be simpler for some transformations, especially in one dimension, where finding P(U ≤ u) directly may be more intuitive. However, in multiple dimensions (as with (X, Y) → (V, W)), direct manipulation of the CDF can be quite cumbersome due to the need to account for multi-dimensional integration limits and piecewise conditions.
A subtle pitfall of the Jacobian approach is ensuring you correctly identify the support in the transformed space. Sometimes it’s easy to overlook restrictions that come from the original variable ranges. Meanwhile, the CDF method can be prone to mistakes if you incorrectly set up the integrals or forget to handle boundary cases. Choosing the right method depends on which approach yields simpler integral boundaries and fewer opportunities for error.
What if we defined W differently, say W = Y/X? Would that change the independence arguments?
Defining W = Y/X is merely an alternate ratio. The essential shape of the distribution changes, but the dependence structure with V remains. If we let V = X + Y and W = Y/X, we can perform the same sort of derivation for the joint density. We’d find that the region of valid (V, W) values again depends intricately on both variables and does not factorize into separate marginal distributions.
Hence, just flipping the ratio from X/Y to Y/X does not restore independence. The sum of two uniform(0,1) variables strongly correlates with the ratio in either form because knowing the sum restricts how large or small the ratio can be. The ratio also restricts the possible sum in nontrivial ways (e.g., if W is very large, that suggests Y >> X and influences the range of V). Therefore, independence does not hold under either W = X/Y or W = Y/X.
Could we use V = X + Y and U = X / (X + Y) instead of V = X + Y and W = X / Y? Would that result in a simpler analysis?
Sometimes defining U = X / (X + Y) makes it easier to see certain properties of the distribution. In fact, for exponential random variables, this transformation famously leads to independence between X + Y and X / (X + Y). However, when X and Y are uniform(0,1), that neat independence property does not emerge.
Still, V = X + Y, U = X / (X + Y) can be simpler to work with in some respects. For example, U automatically remains in (0,1) when X and Y are nonnegative, so you don’t have the unbounded ratio concerns that appear with X/Y or Y/X. You’d do a Jacobian-based transformation similarly:
X = V·U
Y = V·(1 - U)
with the corresponding Jacobian. However, as with the ratio, you must carefully handle constraints like 0 < X < 1 and 0 < Y < 1, which become 0 < V·U < 1 and 0 < V·(1 - U) < 1. You would still discover that V and U are not independent. The essential challenge remains the same: the joint support is complicated, and the joint density does not factorize.
How might the distribution change if X or Y are no longer strictly uniform(0,1) but truncated at a different interval or come from a different distribution altogether?
If X or Y have a different distribution, say they are uniform(a, b) instead of uniform(0, 1) or have some arbitrary continuous distribution, the step-by-step method to get the joint density f_{V,W}(v,w) remains largely the same in principle:
Define the transformations V = g(X, Y), W = h(X, Y).
Solve for X, Y in terms of V, W.
Compute the Jacobian determinant.
Identify the support region in (v, w).
Plug in the correct joint density f_{X,Y}(x,y) of the original variables.
However, the functional forms, bounds, and details of the resulting joint density will generally be more complex. Also, the question of independence becomes even more variable-dependent. Different distributions can yield surprising (in)dependencies. Always check if the sum and ratio transformations are well-defined over the entire domain (e.g., if Y can be zero, ratio-based transformations require special care or might be disallowed).
An additional pitfall is to remember that if (X, Y) are not independent originally, f_{X,Y}(x,y) is not the product of marginals. You need the correct joint pdf. Failing to use the correct joint distribution can lead to incorrect or incomplete results in your transformed space.
In real-world scenarios, how might one validate that the derived joint density is correct?
Validation typically involves a combination of theoretical checks and empirical methods:
Theoretical checks:
Ensure that the derived joint pdf integrates to 1 over the entire support.
Verify that the pdf is nonnegative over that domain.
Check if simpler integrals (like finding marginals) recover known results (e.g., the distribution of X+Y is well-known for uniform(0,1)).
Simulation-based checks:
Generate a large sample of (X, Y) in (0,1) with the known distribution (independent uniforms).
Transform them to (V, W) = (X+Y, X/Y).
Estimate the empirical density of (V, W) using, for instance, a 2D histogram or kernel density estimation.
Compare the empirical distribution with the theoretically derived pdf.
Discrepancies can arise from mistakes in boundary constraints, missing pieces in the piecewise definition, or sign errors in the Jacobian. Sometimes, small deviations in a simulation might be due to sampling variability, so it’s helpful to increase the sample size or run multiple replications to see if the discrepancy is systematic or just random.
Are there special transformations that might accidentally fail the one-to-one requirement, and how do we handle that?
Yes, certain transformations can be many-to-one. For instance, if you define a transformation T(X, Y) that “folds” the plane onto itself (like taking an absolute value or a squared term), the inverse mapping might not be unique. In such cases, you have to split the domain into regions, each mapping to the same point under T. Then you sum the contributions from each region to get the final pdf.
If we used a transformation that lumps together multiple (x, y) points onto the same (v, w) value without adjusting for those overlapping regions, the Jacobian formula alone would give an incomplete or incorrect density. Always confirm that the transformation you choose is invertible within the region of interest. If it’s not strictly invertible, handle each subregion carefully to account for all the (x, y) solutions that map to a single (v, w).
What if we re-center X and Y to be uniform(-0.5, 0.5) and then define V and W in a similar manner? Could that cause additional subtleties?
Shifting the domain to allow negative values introduces new complexities for transformations involving ratios. Specifically, if Y can be negative, the ratio X/Y can pass through both positive and negative values, and it might be undefined at Y=0 if that remains in the support. Moreover, the sum V = X + Y can take positive or negative values. The region for (v, w) then spans multiple quadrants depending on the signs of X and Y.
You also need to handle sign-based constraints carefully. For example, if X and Y can each be negative, you must check whether W = X/Y is positive or negative and how that ties in with the sum V. The one-to-one nature of the transformation might hold in certain subregions (like X>0, Y>0) but differ in subregions where X>0, Y<0, etc. You might have to break down the domain into up to four subregions to capture all sign combinations. Each subregion can have a distinct mapping for (V, W) and potentially a different expression for the Jacobian. This piecewise approach is easy to mishandle if you are not systematic.
When done correctly, the procedure is the same, but the bounding conditions become more elaborate. Additionally, one must be mindful of sign flipping in the Jacobian over different subregions, as that can affect the absolute value. All these nuances emphasize the importance of a careful domain analysis whenever negative values are possible.