Functions, Random Variables, and Why LLMs Are Not Deterministic Systems

In the United States, by around 8th grade (ages 13–14), students are introduced to the concept of a function. This idea becomes foundational across nearly all of mathematics, including algebra, trigonometry, Euclidean geometry, calculus, differential equations, linear algebra, and abstract algebra.

A standard definition of a function is:

A function is a relation between two sets that assigns each element of the first set to exactly one element of the second set.

More formally, if:

\[f : A \rightarrow B\]

then for every:

\[x \in A\]

there exists exactly one:

\[y \in B\]

such that:

\[f(x) = y\]

This “exactly one output per input” property is the key structural constraint that defines a function.

A Critical Asymmetry

It is important to emphasize a subtle but critical asymmetry:

  • Multiple inputs may map to the same output (many-to-one is allowed)
  • A single input may not map to multiple outputs (one-to-many is not allowed)

So it is perfectly valid that:

\[f(x_1) = y\]

and

\[f(x_2) = y\]

with

\[x_1 \neq x_2\]

However, it is not valid for a function that:

\[f(x) = y_1\]

and

\[f(x) = y_2\]

with

\[y_1 \neq y_2\]

This deterministic mapping structure is what makes functions so powerful. It forms the foundation for derivatives, integrals, transformations, and essentially all of continuous mathematics.

Random Variables and Probability

In probability theory, we introduce a different kind of object: the random variable.

A random variable is often defined as a function:

\[X : \Omega \rightarrow \mathbb{R}\]

In this sense, a random variable is still technically a function.

The key difference is that we do not typically focus on the deterministic mapping itself. Instead, we associate the variable with a probability distribution that governs which outcomes are likely.

Rather than asking:

\[X(\omega) = x\]

we ask questions such as:

\[P(X = x)\]

or

\[P(X \in S)\]

for some set $S$.

In other words, probability theory shifts the focus from deterministic outputs to distributions over possible outputs.

Statistics then reverses the direction of inference: we observe samples and attempt to infer properties of the underlying distribution.

Why LLMs Behave Like Stochastic Systems

This distinction becomes especially important when thinking about modern generative AI systems such as Large Language Models (LLMs).

At a computational level, an LLM can be viewed as a function:

\[f_{\theta}(p) \rightarrow \text{distribution over tokens}\]

where:

  • $p$ is the prompt
  • $\theta$ represents the learned model parameters

However, the key point is that the model does not directly produce a single fixed output.

Instead, it produces a probability distribution, and the final response is typically obtained by sampling from that distribution:

\[y \sim P_{\theta}(\cdot \mid p)\]

This means that for the same input prompt $p$, multiple different outputs $y$ can be generated across different runs.

So while the underlying model parameters are deterministic, the output generation process is stochastic.

A Simple Experiment

Consider the prompt:

“Write a paragraph about how to pick your favorite color.”

Running this prompt multiple times will produce different—but still valid—responses.

Each response is typically:

  • Coherent
  • Relevant
  • Grammatically correct

Yet the responses will not necessarily be identical.

This variability is not noise in the traditional sense. It is a direct consequence of sampling from a learned probability distribution over language.

A More Complicated Experiment

The following plot shows how responses vary from LLM to LLM. This graph was generated by using the same prompt 100 times in each of the LLMs listed. The resulting language related qualatative metrics where then calculated. Some models have a “temperature” parameter that can increase or decrease the amount of randomness or creativity in the LLM response. For this experiment, the temperature value was set to 0 for those models that have that parameter available. This would encourage the model to be as non-stochastic as possible when generate responses.

combined plot

In the above we can clearly see two things. First, that across ALL of the LLMs, the response to the prompt was stochastic. Secondly, different models demonstrated different amounts of variance in the quantative metrics of the text that was generated. (Note that this experiment was limited specifically to the english language.)

The prompt used for the experiment above was: “Write a concise paragraph explaining the significance of reproducibility in mach ine learning experiments.” and the system prompt was “You are a helpful assistant.”

The Value and Risk of Stochastic Generation

This stochasticity is central to both the power and the risk of LLMs.

Benefits

Stochastic generation enables:

  • Creative variation
  • Idea generation
  • Exploration of alternative phrasings
  • Discovery of alternative solutions

Risks

At the same time, it introduces:

  • Non-zero probability of incorrect information
  • Inconsistency across runs
  • Difficulty guaranteeing correctness without external verification

Importantly, these risks do not disappear with increased scale or capability.

They are structural properties of the sampling process itself.

Implications for Deployment

In practice, this means LLMs should not be treated as deterministic, truth-preserving systems.

They are better understood as probabilistic generators that require:

  • Validation layers
  • External grounding (retrieval systems, tools, databases, APIs)
  • Testing across distributions of inputs
  • Monitoring for failure modes over time

This is similar in spirit to how humans are evaluated in safety-critical roles.

We do not assume correctness from a single response or test. Instead, we rely on:

  • Repeated evaluation
  • Certification
  • Ongoing oversight
  • Performance monitoring

Closing Thought

LLMs are not simply “right or wrong” machines.

They are systems that generate outputs according to learned probability distributions.

That makes them powerful.

But it also means their outputs should be interpreted as:

Useful suggestions drawn from a distribution, not guaranteed facts.