Probability Theory and Reliability Analysis

Question 1

What type of information on the random variable X is
conveyed by its Cumulative Distribution Function FX(x)?

Answer 1

The cumulative distribution function describes the probability that the random variable is higher or equal to x.

Question 2

How can you compute the mean (or expected value)
value of the probability distribution fX(x)?

Answer 2

The mean of the pdf is calculated by taking the integral of the x * f(x)

Question 3

What is the hazard function of a component? What type of
information does it convey?

Answer 3

The hazard function desribes the failure rate at time t. f(t) / (1-F(t)

Question 4

What type of stochastic event can be represented by the
exponential distribution? What is the parameter that
characterizes this distribution?

Answer 4

The exponential distribution desribes a memoryless with constant failure rate (hazard function).
The parameter lambda determines the failure rate. pdf(t) = lambda * e ^(-lambda * t)

Question 5

What type of stochastic event can be represented by the
Weibull distribution? What are the parameters that
characterize this distribution?

Answer 5

A Weilbull distribution describes a stochastic event, where the failure rate changes over time. It has a paremeter Lambda and a parameter alpha:

F(t) = 1- e^-(lambda*t^alpha)

Question 6

What is the mean-time-to-failure MTTF? How can you
compute it for components with constant failure rate?

Answer 6

The mean time to failure describes when a component is expected to fail.
In components with constant failure rates it is simply the reciprocal of the failure rate.

Question 7

Compute the reliability of a system consisting of identical
components connected in series which are characterized
by constant failure rate λ.

Answer 7

Reliability is defined as R(t) = 1- F(t)
F(t) = 1- e^2*lambda.

All components must function. Hence:
R(all) = Product(R1 * R2 …)

Question 8

Compute the reliability of a system consisting of identical
components connected in parallel which are
characterized by constant failure rate λ

Answer 8

Parallel systems:
All components must fail:
R(t) = 1 - Product (1-R(t))*(1-R(t)….)

lambda = 1/lambda1 + 1/lambda2

Question 9

Show that the reliability of a parallel system of 2
components with constant rate λ1 and λ2 is larger than the
reliability of the serial system with the same components.

Answer 9

For this i would do the Reliability functions and calculate the MTTF!
Or show that one is smaller for all t?