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Trivial Question on Gamma Distribution Function

  1. Sep 11, 2011 #1
    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.
     
  2. jcsd
  3. Sep 11, 2011 #2
    Let X be the standard gamma distribution with parameter [itex]\alpha[/itex]. Then this has pdf

    [tex]p_X(x)=\frac{1}{\Gamma (\alpha)}x^{\alpha-1} e^{-x}[/tex]

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

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

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

    if [itex]Y\sim \Gamma(\alpha,a,\lambda)[/itex], then [itex]Y+k\sim \Gamma(\alpha,a+k,\lambda)[/itex].
     
  4. Sep 11, 2011 #3
    Thank you for your kind help.
     
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