AcademyProbability
Academy
Standard Normal
Level 1 - Math II (Physics) topic page in Probability.
Principle
The standard normal random variable has mean \(0\) and variance \(1\). It provides a common scale for normal probabilities, so tables and calculators usually report its cumulative probability.
Notation
\(Z\sim N(0,1)\)
standard normal random variable with mean 0 and variance 1
\(z\)
a possible value on the standard normal scale
\(f_Z(z)\)
standard normal density
\(\Phi(z)\)
standard normal cumulative distribution function
\(a,b\)
standard normal interval endpoints with a\le b
Method
Step 1: Use the density for area
The standard normal density is symmetric about \(0\). The cumulative distribution function \(\Phi\) stores accumulated area from the far left up to \(z\).
Standard normal density
\[f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2}\]
CDF definition
\[\Phi(z)=P(Z\le z)\]
CDF as area
\[\Phi(z)=\int_{-\infty}^{z}\frac{1}{\sqrt{2\pi}}e^{-t^2/2}\,dt\]
Step 2: Convert intervals to differences of cumulative areas
The area up to \(b\) includes the area up to \(a\). Subtract to isolate the interval.
Area up to b
\[\Phi(b)=P(Z\le b)\]
Area up to a
\[\Phi(a)=P(Z\le a)\]
Subtract included area
\[P(a\le Z\le b)=\Phi(b)-\Phi(a)\]
Rules
Standard normal
\[Z\sim N(0,1)\]
Density
\[f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2}\]
CDF
\[\Phi(z)=P(Z\le z)\]
Symmetry
\[\Phi(z)=1-\Phi(-z)\]
Centre
\[\Phi(0)=\frac{1}{2}\]
Interval
\[P(a\le Z\le b)=\Phi(b)-\Phi(a)\]
Examples
Question
A table gives
\[\Phi(1.20)=0.8849\]
What is \[P(Z\le1.20)\]
?Answer
By definition,
\[\Phi(1.20)=P(Z\le1.20)\]
so the probability is \[0.8849\]
Checks
- \(\Phi(z)\) is cumulative probability, not density.
- The density is \(f_Z(z)\); the CDF is \(\Phi(z)\).
- \(\Phi(0)=1/2\) because the density is symmetric about \(0\).
- An interval probability uses a difference of CDF values.
- Negative inputs can be handled with symmetry when a table lists only positive values.