$X,Y\sim U[0,5]$ are independent. What is the expectation of $Y$, given at least one of the RVs is less than $1$?

Question

How do I approach such a problem without the knowledge of Truncated distributions?

Also, my approach was similar to the person who posted the question: What is wrong with how I am approaching this conditional expectation problem

Answer 1

The only things you need to know about truncated distributions is that a truncated uniform is uniform and the total probability has to equal $1$ for any distribution. The rest works just like in the question you cited.

Answer 2

Just use the definition of conditioning a random variable over an event, followed by the independence of the random variables:

$\qquad\begin{align}\mathsf E(Y\mid X<1\cup Y<1)&=\dfrac{\mathsf E(Y\mathbf 1_{X<1\cup Y<1})}{\mathsf P(X<1\cup Y<1)}\\[1ex]&=\dfrac{\mathsf E(Y\mathbf 1_{X<1\cap Y\geq 1}+Y\mathbf 1_{Y<1})}{\mathsf P((X<1\cap Y\geq 1)\cup(Y<1))}\\[1ex]&=\dfrac{\mathsf P(X<1)\,\mathsf E(Y\mathbf 1_{Y\geq 1})+\mathsf E(Y\mathbf 1_{Y<1})}{\mathsf P(X<1)\,\mathsf P(Y\geq 1)+\mathsf P(Y<1)~}\\[1ex]&~~\vdots\end{align}$