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I'm reading a online book titled Mathematical Statistics Old School which can be found here.

In the section about the Dirichlet Distribution (which begins on page 63 if you would like to look) it says to let $X_1, \ldots, X_K$ be independent random variables, where $X_k \sim Gamma(\alpha_k, 1)$.

Then the $K-1$ vector $\underline{Y}$ defined by $$Y_k = \frac{X_k}{X_1 + \cdots + X_K} , k = 1,\ldots, K-1$$ has a Dirichlet distribution. So $\underline{Y} \sim Dirichlet(\alpha_1, \ldots, \alpha_K)$. It is noted that $Y_1 + \cdots +Y_K = 1$.

The text says the following two things, which I am having trouble understanding:

  1. The marginals $Y_k \sim Beta(\alpha_k, 1- \alpha_k)$
  2. If $X_k \sim Gamma(\alpha_k, \lambda)$, then the definition is the same.

For my first issue, I thought it would be that $$Y_k \sim Beta(\alpha_k, \sum_{i\neq k} \alpha_i)$$ because if $X_1 \sim Gamma(\alpha_1, \lambda)$ and $X_2 \sim Gamma(\alpha_2, \lambda)$ then $$\frac{X_1}{X_1 + X_2} \sim Beta(\alpha_1, \alpha_2).$$

I think clearing this issue up would help me see that if $X_k \sim Gamma(\alpha_k, \lambda)$, then the definition is the same.

EDIT Jan 24 '17: I'm not really sure how my question is a duplication of this question. But I am glad it was pointed out . As pointed out by Xi'an in the comments, my confusion is due to a typo in the text I am reading.

I think it would be more accurate to mark this question as answered instead of as duplicate.

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SOULed_Outt
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    You are right and the book is wrong: the correct result is for instance in Devroye's [Non-Uniform Random Variate Generation](http://amzn.to/2kmdRn9). – Xi'an Jan 24 '17 at 07:28
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    The second issue is straightforward: if $X_k \sim \text{G}(\alpha_k, \lambda)$, then $\lambda X_k \sim \text{G}(\alpha_k, 1)$. You only need to multiply numerator and denominator by $\lambda$ to get back to the original definition. The scale factor has no impact on the ratio. – Xi'an Jan 24 '17 at 07:30

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