IGNOU MSTE 1 2018 - A system having Weibull failure distribution with pdf asi) Com

Question

A system having Weibull failure distribution with pdf as

$f(t) = \frac{1}{\theta }e^\frac{-t}{\theta }t\geq 0$

i) Compute the reliability function of the system,

ii) find the reliability for a 100 days mission for θ = 2,

iii) show that the hazard rate is constant,

iv) find the MTTF, and

v) what is the life of the system if reliability of 0.90 is desired.

17 Feb 2018

Answer :

Word Count : 524

Let's solve the problem step by step. ### Given: The probability density function (PDF) of the system follows a Weibull failure distribution: \[ f(t) = \frac{1}{\theta} e^{-\frac{t}{\theta}}, \quad t \geq 0 \] This is actually an exponential distribution, which is a special case of the Weibull distribution with shape parameter $ k = 1 $. --- ### (i) Compute the Reliability Function The reliability function, $ R(t) $, is the probability that the system survives beyond time $ t $. It is given by: \[ R(t) = P(T > t) = 1 - F(t) \] where $ F(t) $ is the cumulative distribution function (CDF), obtained by integrating the PDF: \[ F(t) = \int_0^t f(x) \,dx = \int_0^t \frac{1}{\theta} e^{-\frac{x}{\theta}} \,dx \] Solving the integral, \[

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