Ratio of gamma values a quarter apart

Question

Evaluate $\Gamma(9/4)$ divided by $\Gamma(1/4)$ .

Short answer

$\tfrac{5}{16}$ . Apply $\Gamma(s+1)=s\,\Gamma(s)$ twice: $\Gamma(\tfrac94)=\tfrac54\,\Gamma(\tfrac54)=\tfrac54\cdot\tfrac14\,\Gamma(\tfrac14)$ , so the ratio is $\tfrac{5}{16}$ .

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Cumulative Distribution Functions
The Chain Rule

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