Find the Distribution of Sample Mean of Size 2 With Uniform Distribution

Question

I know this should be an easy one, and I checked many times, but still cannot believe what I got:

Let $X$ be uniform on $[0,1]$ and $X_1,X_2$ be two random samples from $X$. Find the pdf and cdf of $T=\frac{X_1+X_2}{2}$. Then

\begin{align*} F_{T}(z) &=P(T\leq z)\\ &=P(X_1+X_2 \leq 2z)\\ &=\int_0^1 f_{X_1}(x)P\left(X_2\leq 2z-x\right)dx\\ &=\int_0^1 (2z-x) dx\\ &=2z-\frac{1}{2}\\ f_T(z) &=\int_0^1 2f(x)f(2z-x)dx\\ &=2 \end{align*}

I believe $Z\in [0,1]$, and apparently $F_T(1)\neq 1$. Not sure where I did wrong. Thanks in advance.

Answer 1

Your computation of $F_T(z)$ is valid only when $2z>1$. If $2z <1$ then you have to integrate w.r.t. $x$ from $0$ to $2z$. This is because we have the restriction $X_1 \leq 2T$.