Normal probabilities and quantiles in Java

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Run this example

Complete Getting Started, then run from the repository root:

mvn exec:java -Dexec.mainClass=dev.nm.examples.NormalProbabilityExample

Problem and mathematics

For a standard-normal random variable Z, compute a cumulative probability, a right-tail probability and the 97.5th percentile:

These operations arise in statistical thresholds and uncertainty calculations. Their validity depends on whether a normal model is appropriate for the data.

Walk through the API

The no-argument NormalDistribution constructor creates a distribution with mean zero and variance one. cdf(z) gives cumulative probability, density(z) would give a density, and quantile(p) maps a probability to a value.

The example uses symmetry, normal.cdf(-2), for the right-tail probability at 2. In extreme positive tails, subtracting cdf(z) from one can lose significant digits or round to zero.

Expected result and validation

The CDF reference values are taken from the source test. The executable also checks cdf(quantile(0.975)) against 0.975, to 1e-9, as a functional consistency check. A round trip alone is not an independent accuracy benchmark, which is why reference probabilities are checked too.

Assumptions and mistakes to avoid

A density at one point is not the probability of that point. Quantile inputs are probabilities, not percentages: pass 0.975, not 97.5. For a general normal variable X, standardize with (x-mu)/sigma, where sigma is the standard deviation, not the variance.

Try it

Compute cdf(1.96)-cdf(-1.96) and interpret the resulting central probability. Then examine a far-tail value and compare the symmetry calculation with subtraction from one.

Source and API reference

Adapted from NormalDistributionTest.java, testNormal_0010 and testNormal_0020. See source provenance for the exact NM Dev commit and test path. The full runnable Java source linked above includes imports, construction steps and checks. Browse the NM Dev Javadoc for the classes used here.