AcademyProbability
Academy
Normal Distribution
Level 1 - Math II (Physics) topic page in Probability.
Principle
A normal distribution is a continuous bell-shaped model for quantities made from many small contributions. It is useful for heights, measurement errors, and many small independent disturbances in physics experiments.
Because the model is continuous, exact probabilities are not read from single points. They are areas under a density curve, so interval probabilities come from integrals.
Notation
\(X\)
continuous random variable being modelled
\(\mu\)
mean, the centre of the normal distribution
\(\sigma^2\)
variance, the square of the spread parameter
\(\sigma\)
standard deviation, with sigma positive
\(f_X(x)\)
density of X at the value x
\(a,b\)
interval endpoints with a\le b
Method
Step 1: Use the density as a probability model
The curve height is density, not probability. Probability is area under the curve over an interval.
Density is non-negative
\[f_X(x)\ge0\]
Total area is one
\[\int_{-\infty}^{\infty} f_X(x)\,dx=1\]
Interval probability
\[P(a\le X\le b)=\int_a^b f_X(x)\,dx\]
Exact value probability
\[P(X=a)=\int_a^a f_X(x)\,dx=0\]
Step 2: Read the shape
The mean \(\mu\) is the line of symmetry. Values the same distance to the left and right of \(\mu\) have the same density, so equal-width matching intervals on opposite sides have equal area.
Rules
Total area
\[\int_{-\infty}^{\infty} f_X(x)\,dx=1\]
Interval probability
\[P(a\le X\le b)=\int_a^b f_X(x)\,dx\]
Symmetry
\[f_X(\mu-t)=f_X(\mu+t)\]
Point probability
\[P(X=a)=0\]
Examples
Question
Adult heights in a large population are often roughly bell-shaped. What does the mean represent?
Answer
The mean \(\mu\) represents the centre of the height distribution. The standard deviation \(\sigma\) measures the typical spread of heights around that centre.
Checks
- A normal distribution is continuous, so a single exact value has probability zero.
- The total area under the density curve is \(1\).
- Probabilities are areas under the density, not heights of the curve.
- The mean is the symmetry centre, while the standard deviation measures spread.
- Exact normal probabilities are evaluated using integrals or tables/software based on those integrals.