Let $S$ and $T$ two random indipendent variables with exponential distribution, and let $\mathbb{E}(S)=\alpha,\mathbb{E}(T)=\beta$ .
- 1) Find the distribution of $Y=\min(S,T)$.
$\rightarrow Y\sim Exp(\frac{1}{\alpha}+\frac{1}{\beta})$
- 2) Find the probability of event $\mathbb{P}(S<T)$.
$\rightarrow \mathbb{P}(S<T)=\frac{1}{2}$
Given that these two points I believe they're correct, I have a doubt on the following third point:
- 3) Find $\mathbb{E}(S+T|S>4)$.
$\rightarrow \mathbb{E}(S+T|S>4)=\mathbb{E}(S+T|S>4,T>0)=\mathbb{E}(S|S>4)+\mathbb{E}(T|T>0)=$ $\mathbb{E}(S|S>4)+\mathbb{E}(T)=4+\alpha +\beta$.
Is it correct?
@Francesco Totti (friendly... er pupone?)
The integral is correct but the solution is $\frac{\beta}{\alpha+\beta}$
intermediate result
$\frac{1}{\alpha}\int_0^\infty e^{-s \frac{\alpha+\beta}{\alpha \beta}}ds$
3) it is correct but simply
$\mathbb{E}[S+T|S>4]=\mathbb{E}[S|S>4]+\mathbb{E}[T|S>4]=4+\mathbb{E}[S]+\mathbb{E}[T]$
a) $\mathbb{E}[S|S>4]=4+\mathbb{E}[S]$ follows immediately from lack of memory property
b) $\mathbb{E}[T|S>4]=\mathbb{E}[T]$ follows immediately from independence between S and T