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https://www.geeksforgeeks.org/understanding-the-t-distribution-in-r/
Understanding the t-distribution in R. Last Updated : 24 Feb, 2021. The t-distribution is a type of probability distribution that arises while sampling a normally distributed population when the sample size is small and the standard deviation of the population is unknown. It is also called the Student’s t-distribution.
https://openstax.org/books/introductory-business-statistics/pages/5-3-the-exponential-distribution
P (X > r + t | X > r) = P (X > t) for all r ≥ 0 and t ≥ 0 For example, if five minutes have elapsed since the last customer arrived, then the probability that more than one minute will elapse before the next customer arrives is computed by using r = 5 and t = 1 in the foregoing equation.
https://accendoreliability.com/using-the-exponential-distribution-reliability-function/
$$ \large\displaystyle R(t)={{e}^{-\lambda t}}$$ Given a failure rate, lambda, we can calculate the probability of success over time, t. Cool. An Example. Let’s say we want to know if a new product will survive 850 hours. We have data on 1,650 units that have operated for an average of 400 hours. Overall there have been 145 failures.
http://personal.psu.edu/jol2/course/stat416/notes/chap5.pdf
– Failure rate (hazard rate) function r(t) r(t) = f(t) 1−F(t) – P(X ∈ (t,t+ dt)|X > t) = r(t)dt. – For exponential distribution: r(t) = λ, t > 0. – Failure rate function uniquely determines F(t): F(t) = 1−e− R t 0 r(t)dt. 3
http://websites.umich.edu/~essen/html/byconcept/chapter13.pdf
t m, the cumulative distribution function F(t), and the variance σ2. Next we evaluate E(t), F(t), t m, and σ2 for ideal reactors, so that we have a reference point as to how far our real (i.e., nonideal) reactor is off the norm from an ideal reactor. The functions E(t) and F(t) will be developed for ideal PPRs, CSTRs and laminar flow reactors.
https://stat.ethz.ch/R-manual/R-devel/library/stats/html/TDist.html
Details. The t distribution with df = n degrees of freedom has density . for all real x.It has mean 0 (for n > 1) and variance n/(n-2) (for n > 2).. The general non-central t with parameters (df, Del) = (df, ncp) is defined as the distribution of T(df, Del) := (U + Del) / √(V/df) where U and V are independent random variables, U ~ N(0,1) and V ~ χ^2(df) (see Chisquare).
https://courses.engr.illinois.edu/ece313/fa2013/SectionB/Lectures/lec_20.pdf
some time t is called the reliability R(t) of the component: where F is the distribution function of the component lifetime, X . • The component is assumed to be working properly at time t=0
https://www.ntnu.edu/documents/624876/1277590549/chapt2-1.pdf/4acd3094-e4b9-4bf2-bb20-ec9378d67261
The reliability (survivor) function R(t) The failure rate function z(t) The mean time to failure (MTTF) The mean residual life (MRL) ... Distribution of T I The area under the pdf-curve f (t) is always ... of a new item may, for example, be 15000 hours. When the item is still functioning a˝er 10000 hours, its MRL(10000 hours) may, for example ...
https://ocw.mit.edu/courses/engineering-systems-division/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/lecture-notes/lec13.pdf
Weibull, W., 1951,“A Statistical Distribution Function of Wide Applicability, ... t t o R t e ⎟⎟ ⎠ ⎞ ⎜⎜ ... Hours of Operation Non-Failure Failure 0 0.5 1 1.5 2 2.5 x 10 4 0 2 4 6 8 10 12 UAL Field Data Hours of Operation Non-Failure Failure Field Data on Engine Life
https://cran.r-project.org/web/packages/infer/vignettes/t_test.html
It looks like both of these distributions are centered near 40 hours a week, but the distribution for those with a degree is slightly right skewed. Again, note the warning about missing values—many respondents’ values are missing.
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