What’s Happening

Here’s the thing: An intuitive explanation of transforming random variables correctly.

The post Keeping Probabilities Honest: The Jacobian Adjustment appeared first on Towards Data Science. Introduction Imagine youre modeling customer annoyance from wait times. (we’re not making this up)

Calls arrive randomly, so wait time X follows an Exponential distribution—most waits are short, a few are painfully long.

The Details

Now Id argue that annoyance isnt linear: a 10-minute wait feels more than twice as rough as a 5-minute one. So you decide to model annoyance units as (Y = X²).

Just take the pdf of X, replace x with (\sqrt(y)), and youre done. It looks reasonable—peaked near zero, long tail.

Why This Matters

But what if you actually computed the CDF? Short numpy snippet to confirm this import numpy as np import matplotlib. Stats import expon # CDF of Exponential(1): F(x) = 1 - exp(-x) for x = 0 def cdf_exp(x): return 1 - np.

As AI capabilities expand, we’re seeing more announcements like this reshape the industry.

Key Takeaways

• Exp(-x) # Wrong (naive) pdf for Y = X²: just substitute x = sqrt(y) def wrong_pdf(y): return np.
• Sqrt(y)) # This integrates to 2!

• Quick numerical check of integral from scipy.

• Integrate import quad integral, err = quad(wrong_pdf, 0, np.

The Bottom Line

In this post, well build the intuition, derive the math step by step, see it appear naturally in histogram equalization, visualize the stretching/shrinking empirically, and prove it with simulations. The Intuition To grasp why the Jacobian adjustment is necessary, lets use a tangible analogy: think of a probability distribution as a fixed amount of sand—exactly 1 pound—spread along a number line, where the height of the sand pile at each point represents the probability density.

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