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, \[ F(t) = \left[ -e^{-\frac{x}{\theta}} \right]_0^t \] \[ F(t) = 1 - e^{-\frac{t}{\theta}} \] Thus, the reliability function is: \[ R(t) = 1 - F(t) = e^{-\frac{t}{\theta}}

_____ ________ _________ ____ ____ _____ ______ ____ ____.
_____ _________ __________ _________ _____ _________ ________.
__________ ______ ___ _______ ______ ___ ____ _____ ______ ___ ___ _____.
_________ ___ __________ ________ ______ ____ ________ _________ _____ ___ _________.
______ ___ ____ ________ _________.
___ ____ ____ _______ ________ ______ __________ _____ _________ ___ ___ ____.
____ ______ ______ __________ _________ ________ __________ ____.
___ __________ ____ _________ ______ _______.
_______ ______ ________ ______ _______ _________.
__________ ______ ________ _____ ___ ________.
_____ ___ _______ ________ __________ ____ ______ __________ __________ ________ ___.
_____ ______ _______ ___ ________ ____ ______ _________ _____ _____ ___.
_____ __________ _____ ___ ______ ______.
_______ _____ ______ _________ ______ __________ _____ ______ ____.
___ ________ _______ ____ ______ _________ _______.
___ _____ _______ _______ _________.
__________ ________ _________ ________ ________ _________ ______ ________ ________.
_______ _________ _______ ______ _________.
_______ ___ ____ _______ __________ _______ ________.
____ _______ ________ __________ _______ ________ __________ ____ ______ ____.
________ ___ ______ ________ _____ _____ _____ ___ _______ ___.
_________ ________ __________ _______ ________ ___ _________ ____ _________ ________ _________ ___.
_________ ____ ___ _______ _________ ___ __________.
___ ____ _____ ___ _________ _________ _______ ______ _________ __________.
___ ___ _____ __________ ______ ________ ______ __________ ____ ______.
_________ __________ _______ ________ _____.
_____ _____ ______ _____ ________ __________ _____ ______ _________ ___ ____ __________.
_______ ______ _______ ____ _________ ________ ______ __________ ________ ___ _________ __________.
___ ______ ______ ________ __________ ______.
_________ ______ ______ _____ ___ ___ __________ ______.
______ ___ _______ ____ _________ _________ __________ ___ ______ ___.
_____ ___ ______ ____ __________ _____ _______ ____ ___ _____ __________ ___.
_____ ____ _______ ____ _______ _______ ________ _________ ____ __________ __________ ________.
_______ _____ _________ ___ _____ ______ _______.
_______ _______ ____ __________ ________ _____ _________ __________.
__________ ______ ______ __________ ______ __________ ___ _____.
_____ ____ _______ _________ _____ _________ ________ __________.
____ __________ _______ _____ __________ _________ ________ _____.
________ _____ ______ _____ _____.
_____ ____ _______ __________ _______ _________ ___ ___ _____.
___ _____ ___ ____ ____ _______ _________ __________ __________.
_______ _______ __________ ________ __________.
________ _________ ___ _________ ___ _____ ______ _______.
_________ ____ ______ _________ _________ ________ ______ ___.
________.

IGNOU ACADEMY

Your IGNOU journey, made easier.

Solved Assignments
Study Notes
Guess Papers
Question Papers

Call Us WhatsApp

STUDY • PREPARE • SUCCEED

Skip in 10