 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/discrete-mathematics-continuous-mathematics-and-probability/8/b/i/solution

f_{U, V} (u, v) = 2 e^{- u - 2 v} 1_{{u \geq 0, v \geq 0}} .

For

z \geq 0

, the survival probability of the minimum is

P (Z > z) = P (U > z, V > z) = e^{- z} e^{- 2 z} = e^{- 3 z} .

Thus $Z$ has exponential distribution with rate $3$ , and

f_{Z} (z) = 3 e^{- 3 z} 1_{{z \geq 0}} .

Values of a probability density function at the single endpoint

0

 New to topics? Read the docs here!