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Question

# Find the density of cX when X follows a gamma distribution. Show that only $\lambda$ is affected by such a transformation, which justifies calling $\lambda$ a scale parameter.

Solution

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$Y=cX$

Definition gamma density of $X$:

$f_X(t)=\dfrac{\lambda^{\alpha}t^{\alpha-1}e^{-\lambda t}}{\Gamma(\alpha)}$

with

$\Gamma(x)=\int_0^{+\infty}u^{x-1}e^{-u}du$

Definition cumulative distribution function of $Y$:

$F_Y(y)=P(Y\leq y)$

$Y$ is defined as the $Y=cX$:

$F_Y(y)=P(cX\leq y)$

Let $c>0$. Divide each side of the inequality by $c$:

$F_Y(y)=P\left(X\leq \dfrac{y}{c}\right)$

Use the definition of the cumulative distribution function of $X$:

$F_Y(y)=F_X\left(\dfrac{y}{c}\right)$

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