Let $f(x;p) = p\cdot f_{X_1}(x) + (1-p)\cdot f_{X_2}(x)$ where $X_1 \sim N(1,1)$ and $X_2 \sim N(0,1)$. Based on a sample of size $n = 1$ from $f(x;p)$, derive a one-sided lower $100\lambda\%$ confidence limit of $p$
I tried to find pdf of $f(x;p)$ to see if it belongs to any regular distribution but I failed. I kinda don't know any other way to approach this problem.