Survival Analysis Proof X = min{T, c} - Prove E[H(X)] = F(c)

Question

Suppose $T$ is a non-negative random variable with continuous cumulative distribution function $F$ and cumulative hazard $H$. Let $X =$ min{$T, c$} where $c$ is a constant.

Prove $E[H(X)] = F(c)$

Answer 1

Hint:

$$E[H(X)] = -\int_{0}^{c} \ln(1-F(t))f(t)dt-\int_{c}^{\infty} \ln(1-F(c))f(t)dt.$$ Now substitute $u=1-F(t)$ to prove the given assertion.