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I've learned sum of exponential random variables follows Gamma distribution.

But everywhere I read the parametrization is different. For instance, Wiki describes the relationship, but don't say what their parameters actually mean? Shape, scale, rate, 1/rate?

Expoenetial distribution: $x$~$exp(\lambda)$ $$f(x|\lambda )=\lambda {{e}^{-\lambda x}}$$ $$E[x]=1/ \lambda$$ $$var(x)=1/{{\lambda}^2}$$

Gamma distribution: $\Gamma(\text{shape}=\alpha, \text{scale}=\beta)$ $$f(x|\alpha ,\beta )=\frac{1}{{{\beta }^{\alpha }}}\frac{1}{\Gamma (\alpha )}{{x}^{\alpha -1}}{{e}^{-\frac{x}{\beta }}}$$ $$E[x]=\alpha\beta$$ $$var[x]=\alpha{\beta}^{2}$$

In this setting, what is $\sum\limits_{i=1}^{n}{{{x}_{i}}}$? What would the correct parametrization be? How about extending this to chi-square?

Thanks!

I've learned sum of exponential random variables follows Gamma distribution.

But everywhere I read the parametrization is different. For instance, Wiki describes the relationship, but don't say what their parameters actually mean? Shape, scale, rate, 1/rate?

Exponential distribution: $x$~$exp(\lambda)$ $$f(x|\lambda )=\lambda {{e}^{-\lambda x}}$$ $$E[x]=1/ \lambda$$ $$var(x)=1/{{\lambda}^2}$$

Gamma distribution: $\Gamma(\text{shape}=\alpha, \text{scale}=\beta)$ $$f(x|\alpha ,\beta )=\frac{1}{{{\beta }^{\alpha }}}\frac{1}{\Gamma (\alpha )}{{x}^{\alpha -1}}{{e}^{-\frac{x}{\beta }}}$$ $$E[x]=\alpha\beta$$ $$var[x]=\alpha{\beta}^{2}$$

In this setting, what is $\sum\limits_{i=1}^{n}{{{x}_{i}}}$? What would the correct parametrization be? How about extending this to chi-square?

I've learned sum of exponential random variables follows Gamma distribution.

But everywhere I read the parametrization is different. For instance, Wiki describes the relationship, but don't say what their parameters actually mean? Shape, scale, rate, 1/rate?

Expoenetial distribution: $x$~$exp(\lambda)$ $$f(x|\lambda )=\lambda {{e}^{-\lambda x}}$$ $$E[x]=1/ \lambda$$ $$var(x)=1/\lambda$$

Gamma distribution: $\Gamma(\text{shape}=\alpha, \text{scale}=\beta)$ $$f(x|\alpha ,\beta )=\frac{1}{{{\beta }^{\alpha }}}\frac{1}{\Gamma (\alpha )}{{x}^{\alpha -1}}{{e}^{-\frac{x}{\beta }}}$$ $$E[x]=\alpha\beta$$ $$var[x]=\alpha{\beta}^{2}$$

In this setting, what is $\sum\limits_{i=1}^{n}{{{x}_{i}}}$? What would the correct parametrization be? How about extending this to chi-square?

Thanks!

I've learned sum of exponential random variables follows Gamma distribution.

But everywhere I read the parametrization is different. For instance, Wiki describes the relationship, but don't say what their parameters actually mean? Shape, scale, rate, 1/rate?

Expoenetial distribution: $x$~$exp(\lambda)$ $$f(x|\lambda )=\lambda {{e}^{-\lambda x}}$$ $$E[x]=1/ \lambda$$ $$var(x)=1/{{\lambda}^2}$$

Gamma distribution: $\Gamma(\text{shape}=\alpha, \text{scale}=\beta)$ $$f(x|\alpha ,\beta )=\frac{1}{{{\beta }^{\alpha }}}\frac{1}{\Gamma (\alpha )}{{x}^{\alpha -1}}{{e}^{-\frac{x}{\beta }}}$$ $$E[x]=\alpha\beta$$ $$var[x]=\alpha{\beta}^{2}$$

In this setting, what is $\sum\limits_{i=1}^{n}{{{x}_{i}}}$? What would the correct parametrization be? How about extending this to chi-square?

Thanks!