Probability distribution questions

[indigogirl]

Homework Statement

1. If X, Y, and Z have uniforj density of 1 on unit cube, then find P(X+Y+Z<1)
2. X1, X2, and X3 are independent and normal. Find distribution of Y=(X1^2+X2^2+X3^2)^(1/2)

The Attempt at a Solution

1. I set up a triple integral, but I'm not sure if I got the limits right... P(X+Y+Z<1)=int from 0 to 1, int from y to 1-x, and int from z to 1-x-y... Then I integrated it, but I'm left with the variable z in the answer, which I think is wrong.

2. really not sure

 

[Avodyne]

1. Are you integrating z from z to 1-x-y? This doesn't make sense, does it? If x and y are fixed, the maximum value of z is indeed 1-x-y; what is the minimum value of z?

2. Try something simpler. If x is independent and normal, what is the distribution of y=x^2?

 

[EnumaElish]

1. Easier to visualize the region over which X+Y+Z<1 (in addition to X, Y, Z all being > 0). The ratio of the region to the volume of the unit cube is the answer.

2. You could start by finding the dist. of Y^2=X1^2+X2^2+X3^2. Which distribution describes the sum of squared normal variables?