Sum of two uniforms (dependent)

[steveholy]

Thanks in advance for any feedback.

Homework Statement

X ~ U(a,b) ( X is a Uniform random variable within the interval a-b.)
Y= X+constant

Say, constant = c.

P(X+Y<= z) = ?

Homework Equations

P(X<=x) = (x- a) / (b-a) (The cumulative distribution of a Uniform R.V.)

The Attempt at a Solution

P(X+Y<=z) = P(X+X+c<=z) = P(X<= (z-c)/2) =( (z-c) /2 - a) / (b-a).

Y is dependent on X, that got me confused in being 100% sure that my solution above is correct.

Thanks.

 

[micromass]

Hi steveholy!

The general method looks ok. But:

P(X<=x) = (x- a) / (b-a) (The cumulative distribution of a Uniform R.V.)

This is only true for $x\in [a,b]$. Hence

P(X<= (z-c)/2) =( (z-c) /2 - a) / (b-a).

this is only true for

$$a\leq \frac{z-c}{2}\leq b$$

thus if

$$2a+c\leq z\leq 2b+c$$

if z does not satisfy this, then you'll need to write that the probability is 1 or 0 (depending on the value of z).