Continuous random variable

Gamma distribution

A gamma distributed random variable π‘Œ ∼Gamma(π‘Ž,πœ†) where π‘Ž,πœ† >0 is described by the probability density function prob

π‘“π‘Œ(𝑦)=πœ†π‘ŽΞ“(π‘Ž)π‘¦π‘Žβˆ’1eβˆ’πœ†y

where Ξ“ is the gamma function.

Properties

  1. Expectation: 𝔼⁑[π‘Œ] =π‘Žπœ†
  2. Variance: Var⁑[𝑋] =π‘Žπœ†2
  3. Moments: 𝔼⁑[π‘Œπ‘›] =πœ†βˆ’π‘›Ξ“(π‘Ž+𝑛)Ξ“(π‘Ž) for 𝑛 > βˆ’π‘Ž

Furthermore

  1. The sum of π‘Ž independent exponential random variables is Gamma(π‘Ž,πœ†)
  2. Conjugate prior to Poisson elaborate
  3. A special case is the Chi-squared distribution

Relationship to other distributions

  • By the Central limits theorem for integer 𝑛, Gamma(𝑛,πœ†) ⇝N⁑(π‘›πœ†,π‘›πœ†2) as 𝑛 β†’βˆž.

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