Answer:
To answer these questions, we'll first need to understand some basic concepts related to survival analysis and probability distributions. In this case, we have the survival function S(t) for the lifetime of a battery, which is given by:
S(t) = (5/t)² for t ≥ 0
(a) Mean Residual Life at t = 1:
The mean residual life (MRL) at time t is defined as the expected remaining lifetime at time t, given that the individual has survived up to time t. It can be calculated using the formula:
MRL(t) = ∫[t to ∞] S(u) du / S(t)
Substituting the given survival function:
MRL(1) = ∫[1 to ∞] (5/u)² du / (5/1)²
MRL(1) = ∫[1 to ∞] (25/u²) du / 25
MRL(1) = (1/25) ∫[1 to ∞] (1/u²) du
Now, integrate the function (1/u²) with respect to u from 1 to ∞:
MRL(1) = (1/25) * [-1/u] [from 1 to ∞]
MRL(1) = (1/25) * [0 - (-1/1)]
MRL(1) = (1/25) * 1
MRL(1) = 1/25
So, the mean residual life at t = 1 is 1/25 years.
(b) Hazard Rate at t = 2:
The hazard rate (also called the instantaneous failure rate) at time t is defined as the probability density of failing at time t, given that the individual has survived up to time t. It can be calculated using the formula:
h(t) = -d(log(S(t)))/dt
First, find the survival function S(t):
S(t) = (5/t)²
Now, differentiate S(t) with respect to t:
d(S(t))/dt = d((5/t)²)/dt
d(S(t))/dt = -10/t³
Now, find the hazard rate at t = 2:
h(2) = -d(log(S(2)))/dt
h(2) = -d(log((5/2)²))/dt
h(2) = -d(log(25/4))/dt
h(2) = -d(log(25) - log(4))/dt
h(2) = -d(log(25) - 2*log(2))/dt
h(2) = -d(log(25)) + 2*d(log(2))/dt
Now, differentiate the terms:
h(2) = -(-1/25) + 2*(0)
h(2) = 1/25
So, the hazard rate at t = 2 is 1/25 per year.
(c) Density of T:
The density function (also called the probability density function) of T can be found by taking the derivative of the survival function S(t). In this case, we have already found the survival function S(t):
S(t) = (5/t)²
Now, differentiate S(t) with respect to t to find the density function:
f(t) = d(S(t))/dt
f(t) = -10/t³
So, the density function of T is f(t) = -10/t³ for t ≥ 0.
Explanation: