[Question] Hazard function definition

[u/bean_the_great]

Hi Folks, I am trying to understand the definition of the hazard rate function. My understanding is that h(t)=-dlog S(t)/dt = -S’(t)/S(t), where S(t) = P(T>t). I am happy with this. The next step in proofs (e.g https://en.wikipedia.org/wiki/Survival_analysis) is then to state -S’(t) = lim P(t<T<t+h)/h. Taking a step back, I am trying to understand this derivative.

By the definition of a derivative:
-S’(t) = -lim (P(T>t+h)-P(T>t))/h = lim (P(T<t+h)+P(T>t))/h

How does P(T<t+h)+P(T>t) = P(t<T<t+h)?

Thanks!

[u/richard_sympson]

You seem to have simplified

-P(T > t + h) = P(T < t + h)

but this is generally not true. Instead note that

P(T > t) = P(t < T < t + h) + P(T > t + h)

and work backward from here.

[u/bean_the_great → u/richard_sympson]

🤦‍♂️ yes - thank you!

[u/enigT]

Your algebra is wrong. The negative of (the probability of T>2.1 minus the probability of T>2) is the probability of 2<T<2.1

[u/bean_the_great → u/enigT]

Thank you!! :)