# Statistics: Multivariate Dist/Functions of RVs

1. Nov 27, 2011

### icestone111

1. The problem statement, all variables and given/known data

1) A point (X1,X2,X3) are is chosen at random. X1-3 are uniform distributions across the interval [0,1]
Determine P[(X1-.5)2 + X2-.5)2 + X3-.5)2 ≤ .25]

2) X and Y are random variables with the joint pdf: f(x,y) = 2(x + y) for 0≤x≤y≤1, 0 otherwise.
Find the pdf of Z = X + Y

2. The attempt at a solution

1) So I know the pdf for each one of the random variables is f(x) = 1 when 0≤x≤1
Imagining it in a grid, it would be the volume of the sphere with radius .5 centered at (.5,.5,.5). I am uncertain where to begin applying the pdf of each of the random variables to the graph of the probability needed to be found.

2) I know this has appeared once in another topic, but I can't figure out the bounds of this problem (if I'm not messing up elsewhere).
So the graph envisioned would be the points that lie above the line y=x, bounded by y <= 1, x >=0.
So what I would do from this point is attempt to find the solution for (1-x to 1)∫f(x,z-x)dx = (1-x to 1)∫2zdx = 2z -2z + 2zx = 2zx. What is wrong with this reasoning/where do I approach this problem after this?

Last edited: Nov 27, 2011