Why is the Hazard function not a pdf?

I am trying to understand why the hazard function is not a PDF. For a random variable T, people often define the PDF of this random variable as:


By this definition, the hazard should also be a conditional PDF.

\lambda(t)=\lim_{\delta \to 0} \frac{P(t\leqslant T < t+\delta \mid T\geqslant t)}{\delta}

Seems like the two functions share the same type of definition! They are the limit of a probability. So why is one a PDF and the other not?

I guess is the reason this is not a PDF because the conditioning is not on a single event T=t but rather on T\geqslant? If it were on a single event T=t, would this be a PDF?


The argument of a conditional pdf cannot depend on the conditioning event in any way, shape or form. In
f_{T\mid A}(t\mid A) = \lim_{\delta\to 0} \frac{P\{t < T \leq t+\delta\mid A\}}{\delta}, A can be a fixed event such as \{T>5\} but not something that depends on t such as \{T > t\}.

Another important reason why a hazard function h(t) (or any scalar submultiple thereof) cannot possibly be a pdf is that
\int_0^\infty h(t)\, \mathrm dt = \infty whereas pdfs of lifetimes have more mundane values for their integrals:
\int_0^\infty f_T(t)\, \mathrm dt = 1.

Source : Link , Question Author : DanRoDuq , Answer Author : Dilip Sarwate

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