- #1

Shackleford

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- 2

## Homework Statement

Suppose X ~ Gamma(α, scale = β and that Y = kX with k > 0 a constant. Show that Y ~ Gamma(α, scale = kβ).

## Homework Equations

Gamma distribution, etc.

## The Attempt at a Solution

[itex] λ = \frac{1}{β}

[/itex]

[itex] f(x) = \frac{λ^{α}}{Γ(α)}x^{α-1}e^{-λx}

= {(\frac{1}{β})^{α}}{\frac{1}{Γ(α)}}{(\frac{y}{k})}^{α-1}e^{-λ{(\frac{y}{k})}}

= k{(\frac{1}{kβ})^{α}}{\frac{1}{Γ(α)}}{({y})}^{α-1}e^{-{(\frac{y}{βk})}}

[/itex]

What's with the extra k?