if I sample $n$ $X_i$s from an exponential distirbution; that is $$ X_i \sim exp(1) $$

Then prove that the vector $$ \left ( \frac{X_1}{\sum X_i}, \frac{X_2}{\sum X_i}, \cdots, \frac{X_n}{\sum X_i} \right ) $$

is sampled from the unit simplex in n-dimensions.

  • 1
    $\begingroup$ Just apply the definition: points on the simplex are those with non-negative coordinates summing to unity. Can you demonstrate those facts about this vector? If so, your job is then completed by observing there are $n$ coordinates. $\endgroup$
    – whuber
    Commented Aug 14, 2019 at 16:05
  • $\begingroup$ This question is a specific example of the fact that independent draws from certain gamma distributions can be used to generate Dirichlet-distributed random vectors. $\endgroup$
    – Sycorax
    Commented Aug 14, 2019 at 16:10
  • $\begingroup$ @whuber - I see. so I don't need anything beyond the definition of simplex? because how do I then take into consideration that these are draws from exponential. in fact the definition would apply for any arbitrary distribution that $X_i$ are sampled from. thanks! $\endgroup$
    – asifzuba
    Commented Aug 14, 2019 at 17:56
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    $\begingroup$ Are you sure that your claim is true for an arbitrary distribution? What if the arbitrary distribution has support over the real line? (Answer: stats.stackexchange.com/questions/419751/…) $\endgroup$
    – Sycorax
    Commented Aug 14, 2019 at 18:06
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    $\begingroup$ You are correct about arbitrary distributions--provided they always produce data of the same sign and have zero probability of producing all zeros. The exponential is an example of that, but a Normal (for instance) would not be, nor would a Binomial (for the second reason). $\endgroup$
    – whuber
    Commented Aug 14, 2019 at 18:51


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