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I've found different formulas for the Gamma distribution, one where Gamma(alpha, lambda) has an expected value of alpha/lambda due to the Gamma distribution turning into the following image: https://prnt.sc/lcq5zq. However, in other sites I see Beta being used instead of lambda, where Beta is in the denominator instead (http://prntscr.com/lcq7au), despite being in the same place as lambda in the function. I'd have thought they were merely representative of the numbers used for the Gamma distribution, but these are completely different in representation-- especially since the expected value is now Alpha*beta, instead of alpha/lambda. Can someone explain this for me?

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Both forms are correct as they are just two different parametrizations for the Gamma Distribution. The parameter $\lambda$ wouldn't mean the same thing as the parameter $\beta$. In fact, make the substitution $\lambda = \frac{1}{\beta}$ in your first formula and you will get the second formula. The means also equal using this substitution.

To illustrate a simple example: If I asked you to draw the parametric curve of the unit circle, would you say that the form $$x(t) = \cos(t)$$ $$y(t) = \sin(t)$$

Is different than saying $$x(t) = \sin(t + \pi / 2)$$ $$y(t) = \cos(t + \pi / 2)$$ Both produce the unit circle; one way just draws it counterclockwise while the other draws it clockwise. But both equations draw the same thing.

Searching the Gamma Distribution online will be able to explain better why different parametrizations exist. It depends on the contents of different problems and how the Gamma Distribution is brought up.

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