Distributions of Random Variables

Example 2.14 Covariance from Probability and Statistics Grades. The joint distribution of the probability and statistics grades, X and Y, is given later in Table 2.6 and their marginal distributions are given also later in Table 2.7. What is the covariance of X and F  

To judge the extent of dependence or association between two r.v. s, we need to standardize their covariance. The correlation (or correlation coefficient) is simply the covariance standardized so that its range is [1, 1]. The correlation between X and Y is defined by 

Note that pxy is a unitless quantity, while Cov(X, 7) is not. It follows from the covariance relationship that if X and 7 are independent, then pxr = 0 however, pxr = 0 does 

Example 2.15 Correlation Coefficient from Probability and Statistics Grades. We saw that the covariance between the probability and statistics grades is 0.352 in Example 2.14. Although this tells us that two grades are positively associated, it does not teU us the strength of linear association, since the covariance is nota standardized measure. For this purpose, we calculate the correlation coefficient. Check that Var(Z) = 0.858 and Var(7) = 0.678. Then 

This correlation is not very close to 1, which implies that there is not a strong linear relationship between X and 7. 

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A number of other discrete distributions are listed in Table- 1.1, along with the model on which each is based. Apart from the mentioned discrete distribution of random variable hypergeometrical is also used. The hypergeometric distribution is equivalent to the binomial distribution in sampling from infinite populations. For finite populations, the binomial distribution presumes replacement of an item before another is drawn whereas the hypergeometric distribution presumes no replacement. 

DISTRIBUTIONS OF RANDOM VARIABLES 31 TABLE 2.4 Means and Variances of Continuous Random Variables... 

ABSTRACT In this paper we consider nncertainties in the distribution of random variables due to small-sample observations. Based on the maximum entropy distribution we assume the first four stochastic moments of a random variable as uncertain stochastic parameters. Their uncertainty is estimated by the bootstrap approach from the initial sample set and later considered in estimating the variation of probabilistic measures. 

In this paper we have presented an approach to model uncertainties in the distributions of random variables by maximum entropy formulations based on the first four stochastic moments. Since these moments can not be estimated exactly for small-sample observations we model them as imcertain parameters. Based on a given set of observations we estimated the uncertainties utilizing the bootstrap method. We observed an almost normal distribution of the mean value, the standard deviation and the skewness but a skewed distribution of the kurtosis. 

In practice, in most cases, a renewal function carmot be expressed analytically. Therefore, it is necessary to determine its value using other methods, e.g. using numerical integration, similar procedures as mentioned in literature, tables as mentioned in specialized literature (Blischke Murthy 1994, 1996, Rigdon Basu 2000), etc. Analytical calculation of the integral is only possible for certain types of distribution of random variables. The majority of random variable distributions require a calculation with the use of numerical methods. 

The Cornish-Fisher expansion is a method to approximate the required quantiles of a distribution of random variable based on its cumulants. Cumulant is an alternative to provide the moment of the distribution. It determines the moment of the distribution. In order to apply the Cornish-Fisher expansion, the cumulants and moments of the exponential distribution are needed and can be found in the Appendix. The Cornish-Fisher expansion is... 

H degrees of freedom, as r is an //-vector random variable. Observe that whatever be the distribution of random variable e (thus of r ), we have the mean... 

Hence the probability cannot be negative, and the probability that the value of X is found somewhere in U equals unity. Moreover, let us adopt the condition that the function /x is (not only integrable, but also) sufficiently small at infinity see (E.l.lOa) below. [On the other hand, we do not require/x to be continuous.] In this manner, the randomness of variable X is quantitatively characterized by the joint probability density f. The probability is distributed according to the integrals (E.1.2). The law (E.1.2) is also called the probability distribution of random variable X The probability (of the event) that the value of X is found in 2> is determined by a well-defined integral over 2>. For a random vector variable (with N > ), the distribution is called multivariate. 

Usually in reliability theory (of technical system) probability distributions of random variable T (time-to-failure) are exponential Exp(k) (in case of constant failure rate), Weibull W k, T ) (with appropriate choice of parameters a variety of failure rate behaviors can be modeled), lognormal Log-N i, a ), gamma E(a, p), etc. (Lewis, 1994, Zio, 2(X)7). 

Due to influences of measurement error, manufacturing, assembly and other factors, there is uncertainty in mechanical components geometry size, material properties parameters (such as elastic modulus, Poisson s ratio), and so on. The uncertainty significantly affects the reliability of the mechanical components. Therefore, it is important to choose appropriate distributions of random variables before wear reliability analysis. The traditional probability theory method is one of common methods which are used to deal with the uncertainty of variables. However, the method of probability and statistics is subject to restrictions of sample size, sampling time and sample conditions. Sometimes, because sample size is too small, it is impossible to obtain the probability density functions of random variables. 

We assume that the case we observe is a stochastic process with time dependence. Examples of applications might be found e.g. in Si et al. (2012), Linden (2000), Smith Lansky (1994), Doksum Hoyland (1992) or Sherif Smith (1980), The generation of Fe particles is operation time dependant. Therefore the application of a diffusion process seems to be perfectly adequate. Due to normal distribution of random variable and its application capabilities, the Brownian motion might be used universally. The application of the Brownian motion can be found in many areas. Standard use is related to modelling with the use of differential... 

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