Characteristic Function¶
Formula¶
\[
\varphi_X(t) = \mathbb{E}[e^{itX}]
\]
Parameters¶
- \(t\): real parameter
- \(i=\sqrt{-1}\)
What it means¶
Fourier transform of the probability distribution of \(X\).
What it's used for¶
- Uniquely identifying distributions.
- Proving convergence in distribution.
Key properties¶
- Always exists
- Determines the distribution uniquely
Common gotchas¶
- Don't confuse with MGF; \(i\) makes it bounded.
- Inversion formulas require regularity conditions.
Example¶
If \(X\) is Bernoulli(0.5), then \( arphi_X(t)=0.5(1+e^{it})\).
How to Compute (Pseudocode)¶
Input: quantities required by the card formula (distribution parameters, samples, or test setup)
Output: card-specific statistic/probability/result
compute any required summary quantities from data or model parameters
apply the card formula or workflow
return the resulting value(s)
Complexity¶
- Time: Depends on whether the card is applied analytically, numerically, or from sample data (common sample-statistic workflows are often linear in sample size)
- Space: Depends on whether summaries are streamed or full samples/tables are materialized
- Assumptions: Exact runtime/memory is method-dependent and driven by the chosen estimator/test/distribution representation