Description
Recover P(xLower<X<=xUpper,yLower<Y<=yUpper) from four explicitly established joint CDF corner values using rectangle inclusion-exclusion; no distribution fitting.
Event probabilities and probability masses lie between zero and one. Marginals, intersections, conditioning events, and endpoint atoms describe different quantities. A probability bound or local consistency check can constrain a model without specifying its complete distribution or dependence structure.
Recover P(xLower<X<=xUpper,yLower<Y<=yUpper) from four explicitly established joint CDF corner values using rectangle inclusion-exclusion; no distribution fitting.
When to use this tool
Use Probability: Joint Outcome Single Rectangle Mass when you need the stated event identity, boundary policy, distribution component, or local feasibility check.
How to use the tool and how it works
- Enter cdfUpperUpper, cdfLowerUpper, cdfUpperLower, cdfLowerLower. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Require four CDF values within zero through one and their coordinatewise order. Compute rectangleMass=cdfUpperUpper-cdfLowerUpper-cdfUpperLower+cdfLowerLower; require it between -1e-12 and 1+1e-12 and return it unchanged inside result.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| cdfUpperUpper | Cdf Upper Upper | Required. Finite number. |
| cdfLowerUpper | Cdf Lower Upper | Required. Finite number. |
| cdfUpperLower | Cdf Upper Lower | Required. Finite number. |
| cdfLowerLower | Cdf Lower Lower | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Object containing the calculated result members described below. | Required. Object. |
| result.rectangleMass | Lower-open, upper-closed rectangle mass; tiny out-of-bound rounding values are retained. | Calculated result member: number. |
{
"input": {
"cdfUpperUpper": 0.9,
"cdfLowerUpper": 0.3,
"cdfUpperLower": 0.4,
"cdfLowerLower": 0.1
},
"output": {
"result": {
"rectangleMass": 0.3
}
}
}Limitations and assumptions
- This is a lower-open, upper-closed rectangle under the supplied CDF convention. Small negative or above-one results within the absolute tolerance are retained, not clamped, and a globally valid bivariate CDF is not established.
- Outputs are nested inside result. Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Signed zero is exported as zero.
Alternative or Complementary methods
Use a full probability table or distribution when a local mass, bound, or identity is insufficient. Specify dependence, conditioning, support order, and endpoint policy before combining terms.
References
Similar or alternative tools
- Binomial Mass Mode Tie Boundary
Return lowerMode=clamp(ceil(a)-1,0,trials), upperMode=min(trials,floor(a)), and tied when a is an integer from one through trials.
- Censored Event Boundary Mass
Distinguish P(X<t) from P(X<=t) using supplied left-limit and endpoint CDF values.