Trivial Question on Gamma Distribution Function

[I_am_learning]

If a random variable X follows Gamma Distribution Function with parameters K and thita, what does (X+k) follow? if K is a constant.
I think, since adding the constant is just like shifting the origin, the nature of the curve remain unchanged. But what about its parameter?
Thanks.

 

[micromass]

Let X be the standard gamma distribution with parameter $\alpha$. Then this has pdf

$$p_X(x)=\frac{1}{\Gamma (\alpha)}x^{\alpha-1} e^{-x}$$

for x>0. Then we define the generalized gamma distribution as $Y=a+\lambda X$. This has pdf

$$P_Y(x)=\frac{1}{\Gamma(\alpha)\lambda^\alpha}(x-a)^{\alpha-1}e^{-(x-a)/\lambda}$$

if x>a. This is the $\Gamma(\alpha,a,\lambda)$-distribution. The standard gamma is $\Gamma(\alpha,0,1)$. So to answer your question:

if $Y\sim \Gamma(\alpha,a,\lambda)$, then $Y+k\sim \Gamma(\alpha,a+k,\lambda)$.

 

[I_am_learning]

Thank you for your kind help.