Proof: Probability density function of the normal distribution
Index:
The Book of Statistical Proofs ▷
Probability Distributions ▷
Univariate continuous distributions ▷
Normal distribution ▷
Probability density function
Metadata: ID: P33 | shortcut: norm-pdf | author: JoramSoch | date: 2020-01-27, 15:15.
Theorem: Let $X$ be a random variable following a normal distribution:
\[\label{eq:norm} X \sim \mathcal{N}(\mu, \sigma^2) \; .\]Then, the probability density function of $X$ is
\[\label{eq:norm-pdf} f_X(x) = \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2 \right] \; .\]Proof: This follows directly from the definition of the normal distribution.
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Sources: Metadata: ID: P33 | shortcut: norm-pdf | author: JoramSoch | date: 2020-01-27, 15:15.