AskSia

Plus

Question

4[5

points] Consider a discrete failure time response

T

, with dis...

Sep 27, 2024

Solution by Steps

step 1

To prove P(T > a_j | T > a_{j-1}) = 1 - h(a_j) , we start with the definition of conditional probability:

P(A | B) = \frac{P(A \cap B)}{P(B)}

. Here,

A

is the event T > a_j and

B

is T > a_{j-1}

step 2

The event T > a_j given T > a_{j-1} implies that if the failure time is greater than

a_{j-1}

, it can either be greater than

a_j

or not. Thus, we can express this as: P(T > a_j | T > a_{j-1}) = 1 - P(T \leq a_j | T > a_{j-1})

step 3

The probability P(T \leq a_j | T > a_{j-1}) can be related to the hazard function

h(a_j)

as P(T \leq a_j | T > a_{j-1}) = h(a_j) . Therefore, we have P(T > a_j | T > a_{j-1}) = 1 - h(a_j)

step 4

For part (b), we need to prove that

S(t) = \prod_{a_j \leq t} [1 - h(a_j)]

. The survival function

S(t)

is defined as S(t) = P(T > t) . We can express this as the product of the probabilities of surviving past each failure time

a_j

up to

t

step 5

Thus, we can write S(t) = P(T > a_1) \cdot P(T > a_2 | T > a_1) \cdots P(T > a_k | T > a_{k-1}) , where

a_k

is the largest failure time less than or equal to

t

. Each term can be expressed as

1 - h(a_j)

. Therefore,

S(t) = \prod_{a_j \leq t} [1 - h(a_j)]

step 6

For part (c), we need to prove that

P(t = a_m) = h(a_m) \prod_{a_j \leq a_{m-1}} (1 - h(a_j))

. The probability of the event

t = a_m

can be expressed using the hazard function and the survival function

step 7

We can express

P(t = a_m)

as the probability of surviving up to

a_m

and then failing at

a_m

: P(t = a_m) = P(T > a_{m-1}) \cdot P(T \leq a_m | T > a_{m-1}) . This gives us

P(t = a_m) = S(a_{m-1}) \cdot h(a_m)

step 8

Substituting

S(a_{m-1}) = \prod_{a_j \leq a_{m-1}} (1 - h(a_j))

, we get

P(t = a_m) = h(a_m) \prod_{a_j \leq a_{m-1}} (1 - h(a_j))

Answer

The proofs for parts (a), (b), and (c) have been established as required.

Key Concept

The hazard function and survival function are key concepts in survival analysis, representing the instantaneous failure rate and the probability of surviving past a certain time, respectively.

Explanation

The proofs demonstrate the relationships between the survival function, hazard function, and the probabilities of failure times in a discrete failure time model.