cumulative distribution function

[bizan]

Hello to everyone and thanks for your time.

I have a variable theta, distributed normally. I have a function V.

I know how to calculate the probability of theta being higher than something Pr(theta>0).

This is the question, how to calculate the probability of the function V evaluated at theta to be bigger than something? Pr(V(theta)>0).

Kind regards, Luis.

 

[BGM]

Just want to say that most of the time the required probability \( \Pr\{V(\Theta) > 0\} \) has no closed-form expression, same as the CDF of normal.

Theoretically, you will need to compute the integral

\( \Pr\{V(\Theta) > 0\} = \int\limits_{\theta: V(\theta) > 0} f(\theta)d\theta \)

where \( f \) is the pdf of the normal random variable \( \Theta \)

So you may need to rely on some numerical method, or estimate it by simulation.

 

[bizan]

Oh, I see now! you integrate over the values that make the function higher than 0. I think I understand the logic now. Thank you very much.



PS: The fact that there might be no closed form expression is ok. I would integrate it numerically in matlab as this is a constraint of a bigger maximization problem.