When these transformations are applied to a

continuous random variable U with mean 0 and variance 1 such that its pdf [f.

Lemma 1 : For a non-negative

continuous random variable X, define Z = aX + b, where a>0,b [greater than or equal to] 0 are constant.

Let the X

continuous random variable with probability function f, H : R [right arrow] R is measurable function on (R, B) and H(X) new random variable.

tau](x) is a

continuous random variable with probability density function [p.

The power of using a normal distribution is that the two summary values--the mean and standard deviations--allow for the specification of each

continuous random variable, or more importantly a grouping of random variables.

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:

Roughly speaking,

continuous random variables are found in studies with morphometry, whereas discrete random variables are more common in stereological studies (because they are based on the counts of points and intercepts).

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].

p][Y], which for

continuous random variables is defined as

They and their contributors cover Ostrowski-type results in certain distribution functions, other Ostrowski-type results and applications for probability density functions (PDFs) trapezoidal type results and applications for PDFs, inequalities for cumulative distribution functions via Gruss-type results, elementary inequalities for variants, inequalities for n-type time differential PDFs including Lebesque norms, and variances and moments of

continuous random variables defined over a finite period.