AcademyProbability
Academy
General Normal
Level 1 - Math II (Physics) topic page in Probability.
Principle
A general normal random variable is obtained by shifting and scaling a standard normal random variable. The location parameter \(\mu\) sets the mean, and the positive scale \(\sigma\) sets the standard deviation.
Notation
\(X\sim N(\mu,\sigma^2)\)
normal random variable with mean mu and variance sigma squared
\(\mu\)
mean and centre of X
\(\sigma\)
standard deviation of X, with sigma positive
\(\sigma^2\)
variance of X
\(Z=(X-\mu)/\sigma\)
standardised version of X
\(\Phi\)
standard normal cumulative distribution function
Method
Step 1: Standardise
Subtract the centre and divide by the standard deviation. This converts the measured variable into standard normal units.
Standardise
\[Z=\frac{X-\mu}{\sigma}\]
Solve for X
\[\sigma Z=X-\mu\]
Shift back
\[X=\sigma Z+\mu\]
Step 2: Derive mean and variance
Use \(E[Z]=0\), \(\operatorname{Var}(Z)=1\), and the fact that \(\mu\) is fixed.
Start from scaled form
\[X=\sigma Z+\mu\]
Expectation
\[E[X]=E[\sigma Z+\mu]\]
Linearity
\[E[X]=\sigma E[Z]+\mu\]
Use E[Z]=0
\[E[X]=\sigma\cdot0+\mu=\mu\]
Variance
\[\operatorname{Var}(X)=\operatorname{Var}(\sigma Z+\mu)\]
Shift does not change variance
\[\operatorname{Var}(X)=\operatorname{Var}(\sigma Z)\]
Scale variance
\[\operatorname{Var}(X)=\sigma^2\operatorname{Var}(Z)\]
Use Var(Z)=1
\[\operatorname{Var}(X)=\sigma^2\]
Rules
Standardisation
\[Z=\frac{X-\mu}{\sigma}\]
Density
\[f_X(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}\]
CDF transformation
\[P(X\le x)=\Phi\left(\frac{x-\mu}{\sigma}\right)\]
Interval transformation
\[P(a\le X\le b)=\Phi\left(\frac{b-\mu}{\sigma}\right)-\Phi\left(\frac{a-\mu}{\sigma}\right)\]
Mean and variance
\[E[X]=\mu,\quad \operatorname{Var}(X)=\sigma^2\]
Examples
Question
A measured length in mm is modelled by
\[X\sim N(50,4)\]
meaning mean \[50\]
mm and standard deviation \[2\]
mm. Standardise \[X=53\]
mm.Answer
The standardised value is
\[z=\frac{53-50}{2}=1.5.\]
Checks
- In \(N(\mu,\sigma^2)\), the second parameter is variance, not standard deviation.
- \(\sigma\) must be positive and has the same units as \(X\).
- Standardising subtracts \(\mu\) before dividing by \(\sigma\).
- Normal probabilities for \(X\) are converted to standard normal probabilities for \(Z\).
- The CDF transformation uses the endpoint \(x\), not an interval width.