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