Some Applications of Dirac\u27s Delta Function in Statistics for More Than One Random Variable

In this paper, we discuss some interesting applications of Dirac\u27s delta function in Statistics. We have tried to extend some of the existing results to the more than one variable case. While doing that, we particularly concentrate on the bivariate case

Some Applications of Dirac\u27s Delta Function in Statistics for More Than One Random Variable

In this paper, we discuss some interesting applications of Dirac\u27s delta function in Statistics. We have tried to extend some of the existing results to the more than one variable case. While doing that, we particularly concentrate on the bivariate case

Limit distributions of random walks on stochastic matrices

Problems similar to Ann. Prob. 22 (1994) 424–430 and J. Appl. Prob. 23 (1986) 1019–1024 are considered here. The limit distribution of the sequence XnXn−1 ··· X1, where (Xn)n≥1 is a sequence of i.i.d. 2 × 2 stochastic matrices with each Xn distributed as μ, is identified here in a number of discrete situations. A general method is presented and it covers the cases when the random components Cn and Dn (not necessarily independent), (Cn, Dn) being the first column of Xn, have the same (or different) Bernoulli distributions. Thus (Cn, Dn) is valued in {0, r}2, where r is a positive real number. If for a given positive real r, with 0 \u3c r ≤ 1 2 , r−1Cn and r−1Dn are each Bernoulli with parameters p1 and p2 respectively, 0 \u3c p1, p2 \u3c 1 (which means Cn ∼ p1δ{r} + (1 − p1)δ{0} and Dn ∼ p2δ{r} + (1 − p2)δ{0}), then it is well known that the weak limit λ of the sequence μn exists whose support is contained in the set of all 2 × 2 rank one stochastic matrices. We show that S(λ), the support of λ, consists of the end points of a countable number of disjoint open intervals and we have calculated the λ-measure of each such point. To the best of our knowledge, these results are new

Characterizations of Exponential Distribution Based On Sample of Size Three

Two characterizations of the exponential distribution based on equalities among order statistics in a random sample of size three are proved. This proves two conjectures stated recently in Arnold and Villasenor [4]

APPLICATION OF SOFT COMPUTING TECHNIQUES OVER HARD COMPUTING TECHNIQUES: A SURVEY

Soft computing is the fusion of different constituent elements. The main aim of this fusion to solve real-world problems, which are not solve by traditional approach that is hard computing. Actually, in our daily life maximum problem having uncertainty and vagueness information. So hard computing fail to solve this problems, because it give exact solution. To overcome this situation soft computing techniques plays a vital role, because it has capability to deal with uncertainty and vagueness and produce approximate result. This paper focuses on application of soft computing techniques over hard computing techniques

Immunomodulatory Role of Ocimum gratissimum and Ascorbic Acid against Nicotine-Induced Murine Peritoneal Macrophages In Vitro

The aim of this present study was to evaluate the immune functions and immune responses in nicotine-induced (10 mM) macrophages and concurrently establish the immunomodulatory role of aqueous extract of Ocimum gratissimum (Ae-Og) and ascorbic acid. In this study, nitrite generations and some phenotype functions by macrophages were studied. Beside that, release of Th1 cytokines (TNF-α, IL-12) and Th2 cytokines (IL-10, TGF-β) was measured by ELISA, and the expression of these cytokines at mRNA level was analyzed by real-time PCR. Ae-Og, at a dose of 10 μg/mL, significantly reduced the nicotine-induced NO generation and iNOSII expression. Similar kinds of response were observed with supplementation of ascorbic acid (0.01 mM). The administration of Ae-Og and ascorbic acid increased the decreased adherence, chemotaxis, phagocytosis, and intracellular killing of bacteria in nicotine-treated macrophages. Ae-Og and ascorbic acid were found to protect the murine peritoneal macrophages through downregulation of Th1 cytokines in nicotine-treated macrophages with concurrent activation of Th2 responses. These findings strongly enhanced our understanding of the molecular mechanism leading to nicotine-induced suppression of immune functions and provide additional rationale for application of anti-inflammatory therapeutic approaches by O. gratissimum and ascorbic acid for different inflammatory disease prevention and treatment during nicotine toxicity