# is this sampling from a simplex?

Simply,

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.

• 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.
– whuber
Commented Aug 14, 2019 at 16:05
• 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.
– Sycorax
Commented Aug 14, 2019 at 16:10
• @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! Commented Aug 14, 2019 at 17:56
• 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/…)
– Sycorax
Commented Aug 14, 2019 at 18:06
• 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).
– whuber
Commented Aug 14, 2019 at 18:51