In-depth guide: cumulative distributions in physics
1) What a CDF means experimentally
A cumulative distribution function (CDF) gives the probability that a random variable falls at or below a threshold
x. In laboratories, that threshold can be a detector cutoff, an energy limit, or a timing window.
For example, F(2 ns)=0.95 means 95% of events arrive within 2 nanoseconds.
2) CDF versus PDF in physical terms
The probability density (PDF) describes “where values concentrate,” while the CDF accumulates those densities.
A PDF peak near 0 indicates many small fluctuations, but the CDF tells how often a limit is met.
If you differentiate a smooth CDF, you recover the PDF, which helps validate modeling choices.
3) Normal CDF for measurement uncertainty
Gaussian noise is common in thermal and electronic systems. With mean μ and standard deviation
σ, the normal CDF estimates compliance probability. If μ=10 and
σ=2, then P(X≤14) is about 0.977, indicating only ~2.3% exceed 14.
4) Exponential CDF for decay and waiting times
Waiting-time processes with a constant rate often follow an exponential model. For rate
λ=0.8 s⁻¹, the probability of observing a decay by t=2 s is
1−e^(−λt)≈0.798. This makes the CDF a direct “completed by time” metric.
5) Poisson and binomial for counting experiments
Photon counts, radioactive emissions, and collision events are often discrete. A Poisson CDF with
μ=5 can estimate the chance of seeing ≤3 counts in a window. Binomial CDFs quantify finite trials,
such as “at most 12 hits out of 20,” useful in thresholded detection and quality checks.
6) Lognormal and Weibull in complex systems
Multiplicative variability can yield lognormal behavior, seen in particle sizes and some turbulence metrics.
Weibull models are popular in fracture and lifetime analysis because the shape parameter
k controls early-failure versus wear-out regimes. A larger k produces a steeper rise in the CDF.
7) Parameter and unit discipline
Keep x and parameters in consistent units, especially for time and energy thresholds.
Ensure constraints are met: σ>0, λ>0, and bounds satisfy a<b.
For discrete models, interpret x as a count cutoff, not a continuous value.
8) Practical checks and reporting
Always verify tails by comparing P(X≤x) and P(X≥x), which should sum to 1 for continuous models,
up to rounding. Report thresholds, parameters, and assumptions alongside results. Exporting CSV and PDF helps preserve
the exact model used for peer review and reproducibility.
FAQs
1) What does P(X ≥ x) represent here?
It is the upper-tail probability, meaning outcomes at or above the threshold. For continuous models it equals 1 − P(X ≤ x). For discrete counts it is P(X ≥ ceil(x)), which reflects integer outcomes.
2) Which distribution should I choose for sensor noise?
If noise is symmetric and many small effects add up, normal is a strong default. If values are strictly positive and skewed, consider lognormal. For time-to-event behavior, exponential can be appropriate.
3) Why does the calculator show “invalid range” sometimes?
Some models require constraints: σ must be positive, λ must be positive, and uniform needs a < b. Lognormal and exponential require x ≥ 0. Discrete models need n ≥ 0 and 0 ≤ p ≤ 1.
4) How accurate is the normal CDF computation?
It uses a standard numerical approximation to the error function. For typical engineering ranges it is very close to library values. Extreme z-scores may show small rounding differences, which is expected.
5) Can I use non-integer x for Poisson or binomial?
You can, but interpretation changes. The calculator uses the integer cutoff floor(x) for P(X ≤ x). For tail mode it uses ceil(x) in effect. For clarity, enter integer thresholds when analyzing counts.
6) How do I interpret results near 0 or 1?
Probabilities near 0 suggest the threshold is rarely met; near 1 suggests it is almost always met. In design work, these extremes often signal a conservative or risky cutoff, depending on direction.
7) What should I include when exporting results?
Record the distribution name, parameters, x, and which mode you selected. If you are comparing datasets, also note the measurement units and the time window or sample size used to estimate parameters.