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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 $t=0$?

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

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