is for Reliability Engineering and Reliability Theory that describes the probability of a system to complete its expected function during an interval of time. The engineering is an area focused on optimising the reliability or the probability of successful functioning, of systems such as graphs, percolations and mechanical systems such as airplanes. Originally a tool for insurance companies but nowadays more general interdisciplinary research area.
Questions tagged [reliability]
59 questions
12
votes
2 answers
What is the expected number of infected people?
I was recently in an interview and got asked an interesting problem which I want to know the answer to.
Suppose we're in a party with $n$ people. At minute $t=0$, there is $1$ person who has a contagious disease that is transferable by shaking…
AspiringMat
- 2,607
7
votes
4 answers
Can a probability density function not integrate to 1 over its support?
I don’t know much Probability Theory beyond the undergraduate level.
I was trying to model a simple scenario with my family. What is the probability I will develop type 1 diabetes in the following years?
I did some research on the internet, and it…
Melanzio
- 623
5
votes
0 answers
How to find the partial derivatives of the following nested expression?
I want to find the partial derivatives of the expression for $v_3(\boldsymbol{u})$ with respect to $u_1$, $u_2$ and $u_3$ from the expressions below. Here $\Phi$ denotes the cumulative distribution function of the standard normal probability…
4
votes
1 answer
Why is Krippendorf's alpha behaving this way?
Let's say we have the following ordinal data with one subject and five observers:
Q1
1
1
1
1
1
Krippendorf's alpha turns out to be $1$, which means we have perfect agreement (as expected). However, if we introduce a little disagreement in…
Karla
- 65
4
votes
1 answer
Probability that the connection broke down in this grid
See below chart. The yellow blocks are two islands. It's connected by
a grid of cables. There are 16 vertical cables and 9 horizontal cables
and 12 nodes (blue highlighted) in between.
Hurricane strike and each cable has 1/2 probability of…
R.Yeh
- 193
3
votes
2 answers
Expected value of failure time given the component has survived until some time
I have a component with exponential failure function, i.e. the CDF is ($T$ being the time of failure):
$$F(t):= P(T \leq t) = 1 - e^{-\lambda t}$$
I want to find the expected time of failure of the component, given it has survived until time $\tau$,…
ose
- 177
- 7
2
votes
0 answers
Binomial distribution in reliability theory
Do binomial distributions $Bin(n,p)$ always have increasing hazard rate?
xyz
- 1,366
2
votes
2 answers
How can I compute the probability that the component fails between time t and t+δ seconds given it has survived until t?
In this question I sought to find the probability of a component failing in a given interval $[t, t + \delta]$ assuming an exponential distribution ($T$ being the time of failure).
\begin{align*}
F(t)= P(T \leq t) = 1 - e^{-\lambda…
ose
- 177
- 7
2
votes
1 answer
MTTR and MTBF formulas for series system must be associative
On page $614$ of "Introduction to probability models" by Sheldon Ross ($9$th ed.), he describes the MTTR of a series system as a function of the failure rates ($\lambda_i$) and repair rates ($\mu_i$) of the individual components. The expression…
Rohit Pandey
- 7,547
2
votes
1 answer
How compute overall reliability based on two related (dependent) inter-rater reliability coefficients
The Situation
In many disability programs, a physician, psychologist, or other health professional conducts an exam with the claimant. Although it's not easy, a research study could determine an inter-rater reliability estimate1 for these medical2…
2
votes
2 answers
Reliability of an $l$ pod $k$ of $n$ system.
We have $n$ machines that form a system. The system works if $k$ or more of the $n$ machines are running. The machines are hosted in pods. They are evenly distributed across $l$ of the pods (best effort). For example, if there are $3$ pods and $7$…
Rohit Pandey
- 7,547
2
votes
0 answers
Log-Likelihood function of log-Normal distribution with right censored observations and regression
EXERCISE
Find the log-likelihood function for the regression model of log-Normal Distribution considering the right-censored observations
ATTEMPT:
So, we know that the probability density function of log-normal distribution…
Paris K. Patsogiannis
- 603
- 3
- 17
2
votes
0 answers
Wilson Confidence Interval for binomial proportion, 99.9% level yields strange results for p=0.5
Fellow Stackers,
I have a formal customer requirement for a mechanism to have a failure rate of 0.1% or less. After some study, I found various methods for estimating the confidence interval of a binomial proportion and settled on the Wilson (1927)…
K Neff
- 29
1
vote
1 answer
Convex order stochastic dominance for nonnegative integer valued random variables
Let $X$ and $Y$ be nonnegative integer valued random variables (with same mean). It is customary to define convex order stochastic dominance for such variables (denoted $Xj)\leq \sum_{j\geq n}\mathbb{P}(Y>j)$ for…
xyz
- 1,366
1
vote
2 answers
How can I compute the probability that the component fails between time $t$ and $t + \delta$ seconds?
If I have a system which has an exponential failure rate distribution, i.e.
\begin{align*}
F(t) := P(T \leq t) = 1 - e^{-\lambda t}
\end{align*}
where $T$ is the time of failure and $t$ is measured in seconds.
How can I compute the probability that…
ose
- 177
- 7