Take the 2-minute tour ×

I want to solve $P(e^{X_1}+Ke^{X_2}<c) = \alpha$ for $K$, where $c$ is a constant and $X_1,X_2\sim N(0,1)$. To find $P(e^{X_1}+Ke^{X_2}<c)$, I can write it as an expectation $E(I(e^{X_1}+Ke^{X_2}<c))$,where $I()$ is an indicator function then do a Monte Carlo simulation. How can I determine K?

share|improve this question
Did you want to add that X1 and X2 are independent? Is there some statistical test that led you to the function of X1 and x2 that you have? –  Michael Chernick Jul 12 '12 at 16:02
@MichaelChernick, yes $X_1,X_2$ are independent –  Vaolter Jul 13 '12 at 8:50
Where did the function come from? –  Michael Chernick Jul 13 '12 at 9:40

1 Answer 1

As long as you can evaluate $P(...) = f(K)$ for any fixed value of $K$, which you suggest doing by Monte-Carlo, you now have a 1-dimensional root-finding problem, e.g. find the value of $K$ such that $f(K)=\alpha$, where $\alpha$ and $f$ are known.

Numerically, such problems are typically solved by Newton's method http://en.wikipedia.org/wiki/Newton's_method for well-behaving $f$ (need existence of and ability to compute $f'(K)$, which you have) or by the Secant or Bisection methods http://en.wikipedia.org/wiki/Bisection_method and http://en.wikipedia.org/wiki/Secant_method.

share|improve this answer

Your Answer


