Calculator
Example data table
Discrete distribution example for X with normalized probabilities.
| x | P(X = x) | Context |
|---|---|---|
| 0 | 0.20 | Low-energy state |
| 1 | 0.35 | Typical excitation |
| 2 | 0.30 | Higher event count |
| 4 | 0.15 | Rare burst outcome |
Choose “Discrete x,p pairs” and paste the first two columns.
Formula used
- Moment generating function: M(t) = E[e^{tX}].
- Cumulant generating function: K(t) = ln M(t).
- Discrete expectation: M(t) = Σ_i p_i e^{t x_i}.
- Empirical estimate: M(t) ≈ (1/n) Σ_j e^{t x_j}.
- Selected closed forms: Normal: exp(μt + ½σ²t²), Poisson: exp(λ(e^t − 1)), Bernoulli: (1−p)+p e^t.
How to use this calculator
- Select an input mode: built-in model, discrete pairs, or sample data.
- Enter your t value to evaluate M(t).
- Optionally set a t range to generate an MGF table.
- Click Compute MGF to display results above the form.
- Export tables using Download CSV or Download PDF.
Moment generating function analysis
1) What an MGF represents
The moment generating function (MGF) packages an entire probability distribution into a single function, M(t)=E[e^{tX}]. In physics, it is used to summarize fluctuations of energy, particle counts, waiting times, and measurement noise. When M(t) exists near t=0, its derivatives generate raw moments.
2) Linking moments to derivatives
Raw moments follow m_n = E[X^n] = M^{(n)}(0). This makes MGFs useful for turning complicated expectations into calculus. The calculator reports m1 through m4 and also provides central-shape statistics: variance, skewness, and excess kurtosis, which are common in experimental uncertainty reports.
3) Cumulants via the log MGF
The cumulant generating function K(t)=ln M(t) is central in statistical physics because cumulants add under independent sums. For example, if X and Y are independent, then K_{X+Y}(t)=K_X(t)+K_Y(t). This property supports error propagation and aggregation of independent contributions.
4) Normal model data and Gaussian noise
For a normal variable, M(t)=exp(μt+½σ²t²). This form implies all cumulants beyond second order are zero, consistent with symmetric Gaussian noise. In practice, if your empirical skewness is near 0 and excess kurtosis near 0, a normal approximation can be reasonable over moderate t values.
5) Exponential waiting times and domain limits
Exponential variables model memoryless lifetimes and Poisson-process waiting times. The MGF is λ/(λ−t), which only exists for t<λ. If you request a t beyond the domain, the MGF diverges, and the calculator will flag this. Domain checks prevent misleading finite results.
6) Discrete distributions from measured states
Many experiments produce discrete outcomes, such as energy levels, count bins, or quantized sensor states. For discrete support x_i with probabilities p_i, M(t)=Σ p_i e^{t x_i}. The x,p mode normalizes probabilities automatically so you can paste a histogram-like table directly.
7) Using the MGF curve table
The MGF curve over a t range helps you diagnose stability and tail behavior. Heavy-tailed samples often grow rapidly for positive t, while bounded distributions grow smoothly. The table includes K(t)=ln M(t), which is often more numerically stable when M(t) spans large magnitudes.
8) Reporting and exporting results
For professional reporting, include the data mode, parameter values, the chosen t, and the generated moment estimates. When comparing datasets, use consistent t ranges and step counts. Export CSV for analysis pipelines and PDF for lab notes, technical appendices, or stakeholder sharing.
FAQs
1) What is a good choice of t?
Start near 0, such as 0.1 to 0.5, then expand cautiously. Large |t| can amplify numerical error, especially for heavy tails. If M(t) explodes, reduce |t| or tighten your model parameters.
2) Why does the exponential mode warn about t?
The exponential MGF exists only for t<λ. When t approaches λ from below, M(t) becomes very large. If t≥λ, the expectation diverges, so the calculator marks the result as invalid.
3) Should probabilities in x,p pairs sum to 1?
They should, but you do not need to pre-normalize. The calculator rescales nonnegative p values so they sum to 1. Negative probabilities are ignored to prevent invalid expectations.
4) How are moments computed from samples?
Sample mode uses empirical expectations: M(t) is the average of exp(tX) over your values. Raw moments m1–m4 and shape statistics are computed from the same sample using standard weighted formulas.
5) What does excess kurtosis tell me?
Excess kurtosis compares tail heaviness to a normal reference. Values near 0 indicate Gaussian-like tails. Positive values indicate heavier tails and more outliers. Negative values indicate lighter tails, typical for bounded distributions.
6) Is K(t) always better than M(t)?
K(t)=ln M(t) can be numerically easier when M(t) spans many orders of magnitude. However, K(t) requires M(t)>0 and finite. Use both: M(t) for interpretation, K(t) for stability.
7) How many steps should I use for the curve?
Use 21–51 steps for smooth curves without huge tables. For detailed plots, 81–151 steps can help. Very large tables increase export size and can slow PDF generation, so stay under 201 steps.