# Show that Y ~ Gamma()

1. Oct 6, 2015

### Shackleford

1. The problem statement, all variables and given/known data

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

2. Relevant equations

Gamma distribution, etc.

3. The attempt at a solution

$λ = \frac{1}{β}$

$f(x) = \frac{λ^{α}}{Γ(α)}x^{α-1}e^{-λx} = {(\frac{1}{β})^{α}}{\frac{1}{Γ(α)}}{(\frac{y}{k})}^{α-1}e^{-λ{(\frac{y}{k})}} = k{(\frac{1}{kβ})^{α}}{\frac{1}{Γ(α)}}{({y})}^{α-1}e^{-{(\frac{y}{βk})}}$

What's with the extra k?

2. Oct 6, 2015

### Ray Vickson

It comes from $dy = k dx$, so that $\int f_X(x) \, dx = \int f_X(y/x) \, dy/x$.
Thus, $f_Y(y) = (1/k) f_X(y/k)$.
First, tell us what YOU think.

3. Oct 6, 2015

### Shackleford

Eh. It's related to the scale factor and property of the gamma function. In the first line, shouldn't it be y/k and dy/k?

Last edited: Oct 6, 2015
4. Oct 6, 2015

### Ray Vickson

Yes.

Unfortunately, when I was composing the response, I first gave that detailed answer, then changed my mind and went instead with the "non-answer" that would leave the work to you. I thought I had deleted the detailed answer, but the PF editor is tricky: if one is not careful, material one thinks has been deleted turns out to not have been. That is what happened here! I told you more than I wanted to.

5. Oct 6, 2015

### Shackleford

Hm. Are they fixing the bug? Is my addition below correct?

dy=kdx, so that ∫ fX(x)dx = ∫ fX(y/k)dy/k = ∫ fY(y)dy.

I assume that this is the CDF: FY(y)=(1/k)FX(y/k).[/sub]

Last edited: Oct 6, 2015