Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given that $X ,Y$ has the following joint pdf: $$f (x, y) = x + y , \qquad 0 \leq x \leq 1, \quad 0\leq y \leq 1$$ Let $Z = X+Y$. I need help in finding the pdf $f(z)$ of $Z$.

EDIT [by Srivatsan]: Changed the definition of $Z$ from $Z = x+y$ to $Z = X+Y$ in the title and the post.

share|cite|improve this question
You must have meant $Z=X+Y$ rather than $Z=x+y$. – Michael Hardy Dec 6 '11 at 20:18
Hint: Let $0 \leq \alpha \leq 1$ be a fixed number. Compute $P\{Z \leq \alpha\}$. Repeat for $P\{Z \leq \beta\}$ where $1 \leq \beta \leq 2$. You will need to do double integrals to find these probabilities. Draw pictures to help you figure out the limits of the double integrals. Then find $f_Z(z)$ by differentiating $F_Z(z)$. – Dilip Sarwate Dec 6 '11 at 20:20
up vote 1 down vote accepted

Notice that the joint pdf corresponds to a measure (using Iverson bracket): $$ \mathrm{d} F_{X,Y}(x,y) = (x+y) [ 0 \leq x \leq 1] [0 \leq y \leq 1] \mathrm{d} x \mathrm{d} y $$ Let's perform a change of variables $z=x+y$ and $ w = \frac{y-x}{2}$. The corresponding Jacobian equals 1, and we get: $$ \mathrm{d} F_{X,Y}(x(z,w),y(z,w)) = z \left[ 0 \leq z \leq 2, |w| \leq \min\left( \frac{z}{2}, 1-\frac{z}{2}\right) \right] \mathrm{d} z \mathrm{d} w $$ It is now easy to integrate over $w$, giving you the marginal measure: $$ \mathrm{d} F_Z(z) = z \min\left( z, 2-z \right) \left[ 0 \leq z \leq 2 \right] \,\mathrm{d} z = f_Z(z) \mathrm{d} z $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.