# Derivative of expected value over indicator function

Let $$X$$ be a scalar random variable and $$g,h$$ some continuously differentiable functions. When we study

$$m(t)= E[ \mathbb{1}\{ g(X)+th(X) < 0 \} ]$$

is there a way to calculate the derivative $$\partial m(t)/ \partial t$$ at $$t=0$$?

• Not in general. You can at most to find an expression for $P(g(X)+th(X)<0)$ and differentiate. – drhab Mar 5 at 12:37