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Just want to clarify whether the following is correct: If gamma(a,b) ,then exp(a/b)?
where a,b are parameters for gamma and a/b is the parameter for exp
for example, gamma(1,2)=exp(1/2) Is this true for every a,b>0? Thank you!
1 Answer
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The PDF for the $\Gamma(\alpha,\lambda)$ distribution is $$ f(x)=\frac{x^{\alpha-1}\lambda^{\alpha}}{\Gamma(\alpha)}e^{-\lambda x}$$ for $x>0$. This is not the PDF for any exponential distribution unless $\alpha=1$.
However, the gamma and exponential distributions are closely related: if $X_1,\dots,X_k$ are independent and exponentially distributed with parameter $\lambda$, then $X_1+\dots+X_k$ is $\Gamma$ distributed with parameters $k$ and $\lambda$.
