In numerous

realistic disciplines such as medicine, commercial and ?nance, amongst others,

modeling and investigating lifespan data is essential. Quite a few lifetime

distributions have been used to model such kind of data. The excellence of the

techniques used in a statistical analysis rest on extremely on the presumed

probability model or distribution. As a consequence of this, substantial effort

has been spent in the progress of huge classes of standard probability

distributions along with relevant statistical methodologies. Nevertheless,

there still remain various significant complications where the real data does

not follow any of the classical or standard probability models.

The Rayleigh

distribution (RD)is named after the British physicist Lord Rayleigh(1842–1919)106,

also known as Baron John William Strutt Rayleigh and Nobel Prize winner in

physics 1904. Consequent to the

exponential law, the Rayleigh distribution is the mainly far and wide renowned

particular case of the Weibull distribution. It comes up through the Weibull

density when the shape parameter is set equivalent to two. Similarly the square

root of a chi-square

random variable with v = 2, that is of an exponential random variable, follows the RD 83.

The RD was firstly association with an obstacle in acoustics, and has been used

in modelling certain features of electronic waves and as the distance

distribution between individuals in a spatial Poisson process.

Most frequently

however it appears as a suitable model in life testing and reliability theory.

Heading for additional particulars on the RD the reader is referred to Johnson

et al. 92, 93. Approximate Maximum

Likelihood Estimator (MLE) of the Scale Parameter of the RD with Censoring

sample was discussed by Balakrishnan. N 9.

In recent times Surles and Padgett 112

investigational the two-parameter generalized Rayleigh distribution (GRD) be

able to used pretty excellently in modeling strength and general lifetime data.

Kundu and Raqab 68 used diverse approaches to assess the parameters of the

generalized Rayleigh on simple data. Tzong-Ru Tsai and Shuo-Jye Wu 116 was

discussed acceptance sampling based on life time data. In 2016, Dey et al. 30

derived interval and point estimates of the scale and location parameters of a

two parameter RD using progressive Type-II censored samples. Recenty (2017)

Murithi et. al 88 estimate the parameters of the two parameter of RD based on

Type II Censored data.

Oxytocin

is mammalian neurohypophysialnonapeptide hormone secreted by the posterior

pituitary gland revealed to performance vital roles in numerous perceiving

tasks. For example oxytocin behaves as a

neuromodulator, and has been shown to be involved in stress, anxiety, trust,

empathy, social recognition, orgasm, parturition, lactation, maternal

behaviors, and mother-child and pair bonding 13,46,52,75, 104, 118

and 124.

Oxytocin

is biotic fluids has been measured by radioimmunoassay, enzyme immunoassay,

high performance liquid chromatography (HPLC), and liquid chromatography (LC)

plus tandem mass spectrometry (MS/MS). In-tube solid-phase micro extraction

(SPME) using an open tubular fused-silica capillary with an inner surface

coating as the SPME device, is a simple method that can be easily coupled with

LC 87.

Oxytocin is the leading choice medication for improving uterine

narrowing after delivery. There

are oxytocin receptors in the uterus, and receptors have

also been placed in mammary, endothelial, and central nervous

tissue as well. The effect of oxytocin on endothelial receptors

produces a calcium dependent vasodilator effect via stimulation of

the nitric oxide pathway 114.

The breakdowns in the oxytocin system may underlie

assured psychiatric or emotional pathologies summarized by Zingg 135. The substantial haemodynamic effect of oxytocin

5u i.v. in healthy pregnant patients during spinal

anaesthesia for Caesarean section has been published 73, 74, 96,

105, 115.

In section 3.2 we

analyze the fuzzy Rayleigh distribution (FRD) model for on-line in-tube

solid-phase microextraction coupled with liquid chromatography-tandem mass

spectrometry (online in tube SPME LC-MS/MS) method via estimate the fuzzy

expected values and fuzzy variance values for salivary excretion of oxytocin.The

MLE method was used to find the parameter of RD model.

Insection 3.3 we discussed the GRD model in the fuzzy environment, and is

used to analyze the effect of 5u i.v. bolus doses of oxytocin. We calculate the

mean and variance of fuzzy generalized Rayleigh distribution (FGRD) for the

cardiac output and stroke volume after administration of oxytocin for different

alpha cuts.Here we extant the

method of Maximum Likelihood Estimation as this technique gives simpler

estimation as compared to the Method of moments and the Local frequency ratio

method of estimation. Now we are estimate the parameter of the RD from which

the sample comes. Let

be a random

sample of n observations from the

Rayleigh population with pdfIn life time

applications, fickleness is not the lone attribute of vagueness. In many ?elds

of application, owing to the fuzziness of environment and the negligence of

observers, it is sometimes impossible to obtain exact annotations of lifetime 38.

The acquired lifetime data may be “contaminated” and wooly most of the time. In

addition, constrained by human being and other wherewithal in experiment,

mainly for novel equipment’s, unusually long-life equipment’s, and

non-mass-production products, for which there is no comparative dependability

information available, more often than not, the lifetime is based upon

subjective evaluation or rough estimate. That leads to the fuzziness of

lifetime data. In the circumstance RD consider with fuzzy rules.

In our model 123, we established the fuzzy

Rayleigh distributionand we are finding the parameter of Rayleigh distribution

through maximum likelihood estimator. The effect of oxytocin illustrated by

finding the fuzzy mean and variance for different alpha valuesusing the fuzzy

Rayleigh distribution.

Now consider the RD

with fuzzy parameter

that is swapped

with

. The probability of a random variable X follows Fuzzy

Rayleigh distribution is denoted by

the fuzzy probability density function of a random

variable

is defined byLet us consider the

trial in Shujitsu University, School of Pharmacy, in Japan 87. to calculate

the salivary secretion of oxytocin, 2mgmL?1oxytocin solution was directed by

four bouquets (containing ca. 1.47 mg of oxytocin) into the adenoidal

caves of 59 male volunteers. Saliva was

collected by rinsing the mouth of each subject with water, followed by the

collection of saliva samples in Salisoft tubes containing

polypropylene-polyethylene sponge (Assist, Tokyo, Japan). After saliva samples were collected into

Salisoft tubes containing polypropylene-polyethylene sponges, followed by ultracentrifugation

with Amicon Ultra to eliminate the proteins. To eradicate salivary interfering

substances such as mucin, the filtrate was extracted with MonoTip C18, a

monolithic silica adsorbent packed into a micro-tip.

The saliva samples were

successfully analyzed without interference peaks using the established in-tube

SPME LC?MS/MS method with MRM mode detection.

The salivary excretion of oxytocin after intranasal oxytocin

administration was shown in the Fig. 3.1.

Based on the above observation sample the

parameterthe parameter of RD by equation (3.8) is

.The

corresponding fuzzy triangular number = 63.917, 65.812, 67.480.

The

corresponding

is In our model 119, we originate the fuzzy

generalized Rayleigh distribution by deliver ample representation of the fuzzy

properties of the generalized Rayleigh distribution. This proposed distribution

is used toanalyze the effect cardiac output for 5u doses of oxytocin by

measuring the fuzzy mean and fuzzy variance value for lower and upper alpha

cuts.Consider the study taken by 72, Women were given spinal anaesthesia

with isobaric bupivacaine (7 or 10 mg) and sufentanil 4 µg with a prophylactic

phenylephrine infusion or a placebo infusion. An i.v. bolus of oxytocin 5u

(Syntocinon, Novartis, and Copenhagen, Denmark) was injected into a rapidly

running i.v. line immediately after delivery. All women had an arterial line

inserted, and LidCOPlus (LiDCO, London, UK) was used for invasive monitoring of

cardiac output (CO), and other haemodynamic effects

such as stroke volume (SV), and systemic vascular resistance

(SVR). This monitor performs a beatbybeat analysis of the arterial pressure

wave to determine CO and other haemodynamic variables which are stored in the

computer. The CO effect after the administration of the medication

oxytocin illustrated in Fig.3.6. From the experiment the

parameters of GRD are

2.6445 and

5.8812 for CO and

10.0750 and

82.0 for SV for the i.v. bolus of oxytocin 5u.The corresponding fuzzy triangular

number for the parameters of CO are 1.8405, 2.6445, 3.4025,

5.0362, 5.8812, 6.6632 and the

– cuts are 1.8405+0.804?, 2.6445, 3.4025-0.758?,

5.0363+0.845?, 5.8812, 6.6632-0.782?. The fuzzy mean values and variance for CO

after administration of oxytocin are presented in Table 3.2. The corresponding

fuzzy triangular number for the parameters of SV are

9.2710, 10.0750, 10.8330,

81.1550, 82.0, 82.7820 and the

– cuts are

1.8405+0.804?,

2.6445, 3.4025-0.758?,In section 3.3 the

parameter for RD was calculated successfully by using MLE. The mean and

variance values are estimated for the unremitting drawing out and concentration

of oxytocin in saliva samples analysis using FRD. Analyzing of fuzzy mean and

variance shows that for lower alpha cuts has increasing expected salivary

excretion than the upper alpha cuts. The FRD model for investigation of

oxytocin analyzed by online in tube SPME LC-MS/MS method is very handy for

drool examples and for impartial assessment of the biological belongings of

oxytocin.

In section 3.4 a

mathematical model using FGRD was successfully established. Using FGRD the

effects of cardiac output was calculated by finding the mean and variance

values of FGRD. The results shows that an bolus of oxytocin 5u produced prominent haemodynamic changes ,

and the mean values and variance values are increasing for the lower alpha cuts

and decreasing for upper alpha cuts.