The Normal (Gaussian) Distribution & Standard Score Math
“Symmetric bell curve, inflection points at mu ± sigma, Z-score transformation, and table integration.”
1.The Gaussian Distribution & Its Symmetries
§ 2.1.The Gaussian distribution N(mu, sigma^2) is the paramount distribution in statistical machine learning, arising naturally as the limiting distribution of sums under the Central Limit Theorem and as the maximum entropy distribution for a specified mean and variance.
Standardization (Z-Transform):
Any general normal variable X ~ N(mu, sigma^2) is converted to the standard normal Z ~ N(0, 1) via the affine transformation:
Z = (X - mu) / sigma.
The cumulative distribution function of Z is denoted Phi(z) = P(Z <= z).
Key Symmetry Identities:
1. Phi(0) = 0.5
2. Phi(-z) = 1 - Phi(z)
3. P(a <= X <= b) = Phi((b - mu)/sigma) - Phi((a - mu)/sigma)
2. Worked Practice Problem: Standard Normal Interval Probability
Use the symmetry identity Phi(-z) = 1 - Phi(z) to evaluate negative Z-scores.
Problem: Model accuracy scores across 1000 cross-validation splits follow a normal distribution with mean mu = 75% and standard deviation sigma = 8%. Given Phi(1.0) = 0.8413 and Phi(2.0) = 0.9772, find the proportion of models scoring between 67% and 91%.
Step 1: Convert boundary values to Standard Normal Z-scores:
For lower bound x1 = 67:
z1 = (67 - 75) / 8 = -8 / 8 = -1.0
For upper bound x2 = 91:
z2 = (91 - 75) / 8 = 16 / 8 = +2.0
Step 2: Express interval probability in terms of CDF Phi(z):
P(67 <= X <= 91) = P(-1.0 <= Z <= 2.0) = Phi(2.0) - Phi(-1.0)
Step 3: Apply the symmetry identity for Phi(-1.0):
Phi(-1.0) = 1 - Phi(1.0) = 1 - 0.8413 = 0.1587
Step 4: Compute final probability:
P = Phi(2.0) - Phi(-1.0) = 0.9772 - 0.1587 = 0.8185 (81.85%).
Conclusion: Approximately 81.85% of the models achieve accuracy scores between 67% and 91%.
Primary Academic References & Exam Sources
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