Conditional Density of an Exponential Given Gamma

Question

Hey Guys i get stuck in this problem which says:

$(T_n)_{n>1}$ a succession of iid's R.V's of an $\exp(\lambda)$ and let $T=T_1+T_2+\cdots+T_n$

a) Find the law of $T_1$ given $T$

Obviously i know that $T\sim\operatorname{Gamma}(n,\lambda)$, but if i do the classic approach by definition:

$$f_{T_1\mid T}(t)=\frac{f_{T_1,T}(t)}{f_T(t)}$$

But to get the Joint Distribution, also i understand that because of independence all of $f_{T_{i},T}$(t) would be equal.

Any hints would be appreciated.

Answer 1

Hint: let $X_i \sim E(\lambda)$ and $T=\sum_{i=1}^{n} X_i$

so $$P(X_1<a|T=t)=P(\frac{X_1}{T}<\frac{a}{t}|T=t)=P(\frac{X_1}{T}<\frac{a}{t})$$

note $T$ is complete and sufficient for $\lambda$ and $\frac{X_1}{T}$ does not depend on $\lambda$. so by Basu theorem they are independet.

Basu's_theorem

And it is easy to find

$$\frac{X_1}{T}=\frac{X_1}{X_1 +X_2 +\cdots +X_n}\sim Beta(1,n-1)$$