STA418 Test 2
Consider the following hazard functions
h(t) = θ + βt + γt 2
Find expressions for the
Cumulative hazard function H(t)
The cumulative hazard function is the integral of the hazard
function. It can be expressed as:
t
H(t) = ∫ h(u) du
0
Apply the sum rule:
∫ 𝑓(𝑥) ± 𝑔(𝑥) = ∫ 𝑓(𝑥) + ∫ 𝑔(𝑥)
t
H(t) = ∫ θ + βu + γu2 du
0
𝑡 𝑡 𝑡
H(t) = ∫ 𝜃𝑑𝑢 + ∫ 𝛽𝑢𝑑𝑢 + ∫ 𝛾𝑢2 𝑑𝑢
0 0 0
βt 2 γt 3
H(t) = θt + +
2 3
Survival function S(t)
The survival function can be derived from the hazard function
using the following formula:
t
S(t) = e− ∫−∞ h(u)du
t 2 du
S(t) = e− ∫0 θ+βu+γu
𝑡 𝑡 𝑡 2 𝑑𝑢
S(t) = e− ∫0 𝜃𝑑𝑢−∫0 𝛽𝑢𝑑𝑢−∫0 𝛾𝑢
βt2 γt3
S(t) = e−θt− 2 − 3
Probability density function f(t) of T
The probability density function f(t) of T can be derived from the
Consider the following hazard functions
h(t) = θ + βt + γt 2
Find expressions for the
Cumulative hazard function H(t)
The cumulative hazard function is the integral of the hazard
function. It can be expressed as:
t
H(t) = ∫ h(u) du
0
Apply the sum rule:
∫ 𝑓(𝑥) ± 𝑔(𝑥) = ∫ 𝑓(𝑥) + ∫ 𝑔(𝑥)
t
H(t) = ∫ θ + βu + γu2 du
0
𝑡 𝑡 𝑡
H(t) = ∫ 𝜃𝑑𝑢 + ∫ 𝛽𝑢𝑑𝑢 + ∫ 𝛾𝑢2 𝑑𝑢
0 0 0
βt 2 γt 3
H(t) = θt + +
2 3
Survival function S(t)
The survival function can be derived from the hazard function
using the following formula:
t
S(t) = e− ∫−∞ h(u)du
t 2 du
S(t) = e− ∫0 θ+βu+γu
𝑡 𝑡 𝑡 2 𝑑𝑢
S(t) = e− ∫0 𝜃𝑑𝑢−∫0 𝛽𝑢𝑑𝑢−∫0 𝛾𝑢
βt2 γt3
S(t) = e−θt− 2 − 3
Probability density function f(t) of T
The probability density function f(t) of T can be derived from the
