Let X be a continuous random variable
; then the function
s [member of] [R.sub.+], then [tau](x) is a continuous random variable
Below are the computational formulas for a discrete random variable with pdf [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and for a continuous random variable
with pdf f: R[right arrow]R, respectively:
When these transformations are applied to a continuous random variable
U with mean 0 and variance 1 such that its pdf [f.sub.U]* is symmetric about the origin and cdf [F.sub.U]*, the transformation [T.sub.g,h](U) is obtained, which henceforth will be termed Tukey's g-h generalized distribution.
If X is a continuous random variable
, then its cdf is absolutely continuous, which means that there exists a function f : [??] [right arrow] [??] such that F(x) = [[integral].sup.x.sub.-[infinity]]f(t)dt.
Let [X.sub.1], ..., [X.sub.T] be T continuous random variables
with a density function f([x.sub.1], ..., [x.sub.T]), where f([x.sub.1], ..., [x.sub.T]) > 0 for ([x.sub.1], ..., [x.sub.T]) [member of] [D.sup.T] and [D.sup.T] is defined in a similar way to [D.sup.3].
Continuous random variables
have a normal distribution and follow the central limit theorem, i.e., has an expected population mean [mu] and a standard deviation [delta].
They cover sets, measure, and probability; elementary probability; discrete random variables; continuous random variables
; limit theorems; and random walks.
Among their topics are initial considerations for reliability design, discrete and continuous random variables
, modeling and reliability basics, the Markov analysis of repairable and non-repairable systems, Six Sigma tools for predictive engineering, a case study of updating reliability estimates, and complex high availability system analysis.
Chapters cover both conceptual and theoretical understanding of discrete and continuous random variables
, hypothesis testing, simple regression, nonparametric statistics, and more.
Theorem 2 Let X and Y be continuous random variables
with equal means [[mu].sub.X] = [[mu].sub.Y] and let F and G the corresponding densities.
We also introduce the Conditional Left Tail Expectation, denoted by [CLTE.sub.p][Y], which for continuous random variables
is defined as