Proof: Probability density function of the gamma distribution
Index:
The Book of Statistical Proofs ▷
Probability Distributions ▷
Univariate continuous distributions ▷
Gamma distribution ▷
Probability density function
Metadata: ID: P45 | shortcut: gam-pdf | author: JoramSoch | date: 2020-02-08, 23:41.
Theorem: Let $X$ be a positive random variable following a gamma distribution:
\[\label{eq:gam} X \sim \mathrm{Gam}(a, b) \; .\]Then, the probability density function of $X$ is
\[\label{eq:gam-pdf} f_X(x) = \frac{b^a}{\Gamma(a)} x^{a-1} \exp[-b x] \; .\]Proof: This follows directly from the definition of the gamma distribution.
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Sources: Metadata: ID: P45 | shortcut: gam-pdf | author: JoramSoch | date: 2020-02-08, 23:41.