Let $p=P(X+Y\geq t)$ and $t\in \Bbb R$.
Question: Using the classic Monte Carlo method, find an estimator $p_n$ of $p$ using $F^{-1}_X$ and $F^{-1}_Y$
Attempt: I defined $$Z=X+Y$$ then I expressed $$p=1-P(Z \lt t) =1-\Bbb E(\unicode{x1D7D9}_{z<t}) $$ but I didn't know where to go from here, how am I supposed to use $F^{-1}_X$ and $F^{-1}_Y$ to get the expression of the estimator?
You can do this when $X$ and $Y$ are independent. Here are the steps:
Start with $P=0$. For each $i\in\{1,...,n\}$
follow this well known method and draw two independent uniform random variables $U_i,V_i$ on $[0,1]$
calculate $X_i=F_X^{-1}(U_i)$ and $Y_i=F_X^{-1}(V_i)$
calculate $p_i=\mathbb 1_{\{X_i+Y_i\ge t\}}$ and add it to $P$
when you have reached $i=n$ divide $P$ by $n$
This gives you a Monte Carlo approximation of $p=P(X+Y\ge t)$.
