Central Assumptions for Gaussian (Mild) Randomness
The Gaussian bell curve relies on two central assumptions: independence of events and constant step size. If either assumption is violated—through preferential attachment or wild jumps—the randomness becomes wild and scale-invariant.
THE IDEA
Those Comforting Assumptions
Note the central assumptions we made in the coin-flip game that led to the proto-Gaussian, or mild randomness.
First central assumption: the flips are independent of one another. The coin has no memory. The fact that you got heads or tails on the previous flip does not change the odds of your getting heads or tails on the next one. You do not become a “better” coin flipper over time. If you introduce memory, or skills in flipping, the entire Gaussian business becomes shaky.
Recall our discussions in Chapter 14 on preferential attachment and cumulative advantage. Both theories assert that winning today makes you more likely to win in the future. Therefore, probabilities are dependent on history, and the first central assumption leading to the Gaussian bell curve fails in reality. In games, of course, past winnings are not supposed to translate into an increased probability of future gains—but not so in real life, which is why I worry about teaching probability from games. But when winning leads to more winning, you are far more likely to see forty wins in a row than with a proto-Gaussian.
Second central assumption: no “wild” jump. The step size in the building block of the basic random walk is always known, namely one step. There is no uncertainty as to the size of the step. We did not encounter situations in which the move varied wildly.
Remember that if either of these two central assumptions is not met, your moves (or coin tosses) will not cumulatively lead to the bell curve. Depending on what happens, they can lead to the wild Mandelbrotian-style scale-invariant randomness.
When to use it
Evaluating financial market fluctuations where returns may not be independent or have constant variance.
Modeling social success (e.g., popularity, wealth) where cumulative advantage (preferential attachment) plays a role.
Designing scientific experiments that assume independent and identically distributed (i.i.d.) observations; check for dependencies.
Assessing risk in contexts where rare but extreme events (wild jumps) can dominate outcomes, such as insurance or black swan events.
FROM THEORY TO EVERYDAY LIFE
In practice
Coin-Flip Game
A fair coin is flipped repeatedly. Each flip is independent with a constant step size of ±1. The sum of many flips approximates a Gaussian (bell curve) distribution.
Preferential Attachment and Success
In many real-world scenarios (e.g., wealth accumulation, academic citations), past success increases the chance of future success. This violates the independence assumption, leading to distributions with fat tails (e.g., power laws) rather than a Gaussian.
A MOMENT TO REFLECT
Put it to work.
Can you think of a real-world situation where the assumption of independent events fails? How does that affect the outcomes?
What would happen to the distribution if the step sizes in a random walk were not constant but could be very large?
Why might teaching probability using simple games mislead people about randomness in complex systems?
Often confused with
CONTINUE THE CONVERSATION
Source texts
- The Black Swan The Impact of the Highly Improbable With a new section On Robustness and Fragility (Nassim Nicholas Taleb)