- #1

Hamiltonian

- 296

- 193

- Homework Statement
- If the following joint p.d.f. can be considered for the random variables X, Y, and Z:

$$f(x,y,z) = \begin{cases} 2 & for & 0<x<y<1\ \&\ 0<z<1 \\ 0 & otherwise\end{cases}$$

Evaluate ##\mathbb{P}(2X > Y |1 < 4Z < 3).##

- Relevant Equations
- $$f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)} {f_{Y}(y)}$$

using the equation mentioned under Relevant Equations I can get, $$\mathbb{P}(2X > Y |1 < 4Z < 3) = \frac{\mathbb{P}(2X>Y, 1<4z<3)}{\mathbb{P}(1<4z<3)}$$ I can find the denominator by finding the marginal probability distribution, ##f_{Z}(z)## and then integrating that with bounds 0 to 1. But I am a little confused as to the limits of integration I need to use to find ##f_{Z}(z)## and then there's still the question of what I need to do to find the numerator.

$$f_{Z}(z) = \int_{?}^{?}\int_{?}^{?} f(x,y,z) dx dy$$

Additionally, I wonder if this approach is completely flawed and whether there is a better way to approach this problem.

$$f_{Z}(z) = \int_{?}^{?}\int_{?}^{?} f(x,y,z) dx dy$$

Additionally, I wonder if this approach is completely flawed and whether there is a better way to approach this problem.