Joint Density function Question

Question

The joint probability density function is given as$$f(x,y,z) = kxyz^2$$ where$$ 0<x<1,0<y<1,0<z<2$$

The question asks to find $$P(Z>X+Y)$$

I know we’ll have to find the value of $k$ and I have done that part, but I don’t know how to go forward from there, any help?

Answer 1

You just have to integrate the density over the region where $z>x+y$ so$$\int_0^1\int_0^1\int_{x+y}^2f(x,y,z)dzdydx$$

Answer 2

As this seems like a homework question I won't answer your question completely, but I will give you enough help so that you could figure it out from here. Please ask for further clarification if anything is unclear.

I assume $X$, $Y$ and $Z$ are independent. You will need the marginal distributions of all three variables. Since you managed to determine $k$ I assume you know how to get these. Then consider the following assertion:

You don't care about any particular values of $X$ or $Y$. You simply care that, given any two values, what is the probability that $Z$ is larger? Say $X = x$ and $Y = y$. Then this probability will be $\int_{x+y}^{2} f_Z(z) \mathop{\mathrm{d}z}$. If this does not make sense to you, you should take the time to figure it out.

However, you need to account for the fact that $x$ and $y$ are unknown. You can do this by integrating over every possible $x$ and $y$. This also allows you to take into account that not every $x$ is equally probable. This means you will end up having to evaluate a triple integral.