Conditional probability with exponential distribution

Question

Suppose that $X \sim \mathcal{E}(1.3)$ and $Y \sim \mathcal{E}(1.7)$ are two exponential random variables and define $U := \min\{X, Y\}$.

How do I calculate following values?

• $\mathbb{P}[U > 0.32 \mid X > 0.19]$
• $\mathbb{P}[U > 0.19 \mid X > 0.32]$

Answer 1

HINT

The first one is $$ \begin{split} \mathbb{P}[U > 0.32\mid X > 0.19] &= \mathbb{P}[\min\{X,Y\} > 0.32\mid X > 0.19] \\ &= \mathbb{P}[X > 0.32, Y > 0.32\mid X > 0.19] \end{split} $$

Now use independence and compute. Similarly the second one...