Normal distribution probability

Objective

Having a normal distribution with a mean (μ) and a standard deviation (σ).

What is the probability that a random value will be less than some specific bound?

The probability that the random value is less than the bound is the complement of the standard normal cumulative distribution function (CDF).

The standard normal cumulative distribution function

$$\Phi (x) = \frac{1}{2} \left ( 1 + erf \left ( \frac{x}{\sqrt{2}} \right ) \right );$$

Probability that a random value (x) is greater / less or equal than bound (b) at standard normal distribution

Standard normal distribution - is a normal distribution with mean = 0 and standard deviation = 1.

$$P(x > b) = \Phi(b);$$

$$P(x\leqslant b) = 1 - \Phi(b);$$

$$P(x\leqslant b) = 1 - \frac{1}{2} \left ( 1 + erf \left ( \frac{b}{\sqrt{2}} \right ) \right );$$

Probability at normal distribution with mean and standard deviation

$$P(x\leqslant b) = 1 - \frac{1}{2} \left ( 1 + erf \left ( \frac{b-\mu}{\sigma\sqrt{2}} \right ) \right );$$

$$z = \frac{x-\mu}{\sigma};$$

$$P(x\leqslant a) = 1 - \frac{1}{2} \left ( 1 + erf \left ( \frac{z}{\sqrt{2}} \right ) \right );$$

$$P(x\leqslant a) = \frac{1}{2} \left ( 1 - erf \left ( \frac{z}{\sqrt{2}} \right ) \right );$$

• μ - mean;
• σ - standard deviation;
• z - standard score (z-score);

Error function

$$erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-x^2}dx;$$

$$erf(-x) = -erf(x)$$

Expansion of the integral using the Taylor series

$$erf(x)=\frac{2}{\sqrt{\pi}} \sum_{n=0}^{\infty} \frac{x}{2n+1} \prod_{k=1}^{n} \frac{-x^2}{k};$$

Test data

Error function

z	erf(z)
0	0
0.02	0.022564575
0.04	0.045111106
0.06	0.067621594
0.08	0.090078126
0.1	0.112462916
0.2	0.222702589
0.3	0.328626759
0.4	0.428392355
0.5	0.520499878
0.6	0.603856091
0.7	0.677801194
0.8	0.742100965
0.9	0.796908212
1	0.842700793

Probabilities

μ = 20; σ = 2;

b	P(b), 0.XXXX
19	0.3085
19.5	0.4013
20	0.5
21	0.6915
22	0.8413

Implementation

• NormalDistribution.java
• NormalDistributionTest.java

Links

• https://en.wikipedia.org/wiki/Normal_distribution
• https://en.wikipedia.org/wiki/Cumulative_distribution_function
• https://en.wikipedia.org/wiki/Error_function
• https://en.wikipedia.org/wiki/Standard_score