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In mathematics, Zariski splitting is a splitting of a projective variety of the form
where is a vector bundle of rank .
Formulation as a fibration
Zariski splitting is often described as a fibration
where the vector bundle is the normal bundle to the section. If is a section of that does not meet the zero-section, then is a Zariski-connected variety and the associated fibre is equal to. If does not meet the zero-section, then the fibre over is homeomorphic to the product.
Zariski splitting has as special case the Zariski–Nagata splitting, where is an irreducible variety that is invariant under the action of a finite group of automorphisms of. In this case, the set of degenerations is naturally identified with the associated symmetric space.
Examples
The fibre is called a (complex) flag variety for the group. The fibration is the Segre embedding.
An example of a Zariski splitting is the blowup of a projective variety. Any blowup is a birational self-map of the projective variety. If is the blowup of a projective variety, then the exceptional
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How to derive the distribution of $\frac{(X_1+X_2+X_3+X_4+X_5)}{(\sqrt{5}-X_2-X_3)^{12}}$?
Let $X_1,X_2,X_3,X_4,X_5$ independent and identically distributed random variables that are distributed as $Exp(1)$. Let $T=\frac{(X_1+X_2+X_3+X_4+X_5)}{(\sqrt{5}-X_2-X_3)^{12}}$. Find the distribution of $T$.
What I have tried:
As suggested by one of the answers, $T=\frac{(X_1+…+X_5)(\sqrt{5}-X_2-X_3)^{12}}{(\sqrt{5}+X_2+X_3)^{12}}$.
With respect to this form, I have to find the distribution of $X_1+…+X_5$ and $X_2+X_3$ and subsequently divide them (the distribution of a product is the same as of the summation of the distributions and the denominator of this product is independent of the numerator).
I am not sure if I am on the right track or not. Any comments and suggestions are appreciated.
A:
There are a few things that could be quicker.
Let $X_1$ be the first exponential random variable, $X_2$ be the second exponential random variable, $X_3$ be the third exponential random variable, $X_4$ be the forth exponential random variable, and $X_5$ be the fifth exponential random variable.
Then using the law of the sum of exponential random variables, we have
\begin{align*}
\mathbb{P}(T>t) &= \mathbb{P} \left ( X_1+X_2+X_3+X_4+X_5 < (\sqrt5-X_2-X_3)^{12}t \right ) \\
&=
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