Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Normal distribution ▷ Relationship to standard normal distribution

Theorem: Let $X$ be a random variable following a normal distribution with mean $\mu$ and variance $\sigma^2$:

\[\label{eq:X-norm} X \sim \mathcal{N}(\mu, \sigma^2) \; .\]

Then, the quantity $Z = (X-\mu)/\sigma$ will have a standard normal distribution with mean $0$ and variance $1$:

\[\label{eq:Z-snorm} Z = \frac{X-\mu}{\sigma} \sim \mathcal{N}(0, 1) \; .\]

Proof: The linear transformation theorem for multivariate normal distribution states

\[\label{eq:mvn-ltt} x \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad y = Ax + b \sim \mathcal{N}(A\mu + b, A \Sigma A^\mathrm{T})\]

where $x$ is an $n \times 1$ random vector following a multivariate normal distribution with mean $\mu$ and covariance $\Sigma$, $A$ is an $m \times n$ matrix and $b$ is an $m \times 1$ vector. Note that

\[\label{eq:Z-X} Z = \frac{X-\mu}{\sigma} = \frac{X}{\sigma} - \frac{\mu}{\sigma}\]

is a special case of \eqref{eq:mvn-ltt} with $x = X$, $\mu = \mu$, $\Sigma = \sigma^2$, $A = 1/\sigma$ and $b = \mu/\sigma$. Applying theorem \eqref{eq:mvn-ltt} to $Z$ as a function of $X$, we have

\[\label{eq:mvn-ltt-norm} X \sim \mathcal{N}(\mu, \sigma^2) \quad \Rightarrow \quad Z = \frac{X}{\sigma} - \frac{\mu}{\sigma} \sim \mathcal{N}\left( \frac{\mu}{\sigma} - \frac{\mu}{\sigma}, \frac{1}{\sigma} \cdot \sigma^2 \cdot \frac{1}{\sigma} \right)\]

which results in the distribution:

\[\label{eq:Z-snorm-qed} Z \sim \mathcal{N}(0, 1) \; .\]
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Metadata: ID: P180 | shortcut: norm-snorm3 | author: JoramSoch | date: 2020-10-22, 06:34.