- Thread starter
- #1

#### OhMyMarkov

##### Member

- Mar 5, 2012

- 83

The problem:

$X,Y,Z$ are random variables that are dependent and uniformly-distributed in $[0,1]$, and let $\alpha$ be a given number in $[0,1]$. I am asked to compute the following:

$\text{Pr}(X+Y+Z>\alpha \;\;\; \& \;\;\; X+Y\leq \alpha)$

What I have so far

$f_{X+Y+Z,X+Y}(u,v)=f_{Z,X+Y}(u-v,v)=f_{Z}(u-v)\cdot f_{X+Y}(v)$

(1) Is the above equation correct? I think it stands for discrete RVs but not quite sure for continuous RVs... If it is true, is the following integral correct to compute the desired probability?

$\int _{\alpha} ^{+\infty} \int _{-\infty} ^{\alpha} f_{X+Y+Z,X+Y}(u,v) du\; dv = \int _{\alpha} ^{+\infty} \int _{-\infty} ^{\alpha} f_{Z}(u-v)f_{X+Y}(v) du\; dv=

\int _{\alpha} ^{+\infty} \alpha f_{X+Y}(v) dv$

\int _{\alpha} ^{+\infty} \alpha f_{X+Y}(v) dv$