Find PDF of given random variable $Z=\frac{(X+Y)}{2}$

Question

If $X,Y\sim U(0,1)$
Find the PDF of $Z=\frac{(X+Y)}{2}$.

I know that I can start with CDF of Z, then $P(Z < z)=P\left(\frac{(X+Y)}{2}< z\right)=P(X < 2z-Y) $
But I don't know how to continue with that.

Answer 1

Let $g:\mathbb R\to\mathbb R$ take value $1$ if $x+y\leq 2z$ and value $0$ otherwise.

Then:

$$P(Z\leq z)=\mathbb Eg(X,Y)=\int\int g(x,y)f_{X,Y}(x,y)dxdy$$

Actually this can only be calculated if the joint distribution of $(X,Y)$ is known, but in your question you only mention the marginal distributions. I suspect that $X$ and $Y$ are meant to be independent so that:$$f_{X,Y}(x,y)=f_X(x)f_Y(y)$$

Answer 2

The distribution of a sum is the convolution of the distributions.

$$z \sim U(x;0,1/2) \star U(x;0,1/2) = T(0,1)$$

where $T$ is the triangle function, so that

$$f(z) = \left\{ \begin{aligned} 4z, 0\le z < \frac{1}{2} \\ 1-4z, \frac{1}{2} \le z < 1 \\ 0 , \textrm{otherwise} \end{aligned} \right.$$