Let $X_{1},...,X_n$ be a sample from a continuous distribution with function $F$. Define $U_i = F(X_{i})$ for $i = 1,...,n$. Show that random variables $U_i,...,U_n$ form a sample from the uniform distribution on $[0,1]$
The point where I'm stuck is that I don't know how to work with a random variable as a variable of a cumulative distribution function. Do I have to subsitute $X_i$ by something? Or is there a more elegant way of proving it?
Since $F$ is continuous it has IVP. If $t\in [0,1]$ then there exists $s$ such that $F(s)=t$. Choose the largest such $s$. Then $P\{F(X_i)\leq t\}= P\{X_i\leq s\}=F(s)=t$ $\, \,$ (1). Hence $F(X_i)$ has uniform distribution. Of course $F(X_i),i=1,2,\cdots$ are independent. [To get the first step in (1) verify that $F(X_i)\leq t$ iff $X_i \leq s$].