2.1.5 Hazard Rate Functions Flashcards
(15 cards)
what does a hazard function measure?
It measures the instantaneous rate of failure at time (t), given survival up to time (t).
What is the formula for the hazard rate function?
h(t) = f(t)/S(t), where f(t) is the PDF and S(t) = P(X > t) is the survival function.
What does the hazard rate function represent conceptually?
The probability that the event occurs in the next instant, given that it hasn’t happened yet.
What does the denominator of the hazard function (S(t)) represent?
The probability that the random variable has survived past time
t.
What does the numerator of the hazard function (f(t)) represent?
The probability density that the event occurs exactly at time ( t ).
In life insurance, what is the hazard rate called?
The force of mortality — measures instantaneous death rate. i.e. probability that a person dies within the next instant
What does a high hazard rate mean at time t?
The event is very likely to occur immediately after time ( t ), assuming survival up to that point.
Can the hazard rate ever decrease?
Yes — it depends on the distribution. For some, hazard rate drops over time (e.g. infant mortality model).
What kind of situations use hazard functions outside of insurance?
Reliability engineering, medicine (time to failure or death), risk analysis.
Why is hazard rate not equal to probability?
It’s an instantaneous rate, not a cumulative or total probability.
What does it mean if ( h(t) ) spikes suddenly at time ( t )?
There’s a sudden increase in risk at that time.
How does hazard rate help in modeling real-world events?
It tells us when an event is likely to occur based on survival, not just whether it will.
What’s a key difference between the hazard function and the PDF?
The PDF gives absolute likelihood at a time; hazard rate gives conditional likelihood given the survival function.
Can the hazard rate be greater than 1?
Yes — because it’s a rate, not a probability.
Why is understanding hazard rates useful later in actuarial studies?
They form the foundation of survival models, used extensively in life contingencies and health insurance.
