On the probability of extending an observation Final report, Nov. 1968 - May 1969
Calculation of conditional probabilities for random variables in scheduling spacecraft launching
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Calculation of conditional probabilities for random variables in scheduling spacecraft launching
Cumulative probability distribution of positive random variable from moment generating function, exemplifying exponential and Poisson functions
Optimal rule for decision to stop or continue observation of random variables after observing sequence of variables with continuous distribution function
Random variable method of analyzing propulsion system performance and stage inert weight of two stage rocket vehicles
Histogram method for combining random variables in predicting microbial burdens on spacecraft
Convolution formulas for Cauchy and standard normal random variable distributions derived from geometry of circular symmetric distribution on plane
Algorithm for fast digital computer recursive estimation of mean of random variable
Standard deviation estimates causing operating characteristic curves to be random variables in quality control acceptance sampling plans and methods for computing confidence limits
Photoelectron count of lognormally fading optical signal, discussing noncentral chi square random variable approximation
The possibility of using on-line signal statistics to detect electronic equipment nonlinearities is discussed. The results of an investigation using Gaussian statistics are presented, and a nonlinearity test that uses ratios of the moments of a Gaussian random variable is developed and discussed. An outline for further investigation is presented.
The technical development of a computer program for predicting microbial burden on unmanned planetary spacecraft is outlined. The discussion includes the derivation of the basic analytical equations, the selection of a method for handling several random variables, the macrologic of the computer programs and the validation and verification of the model. The prediction model was developed to (1) supplement the biological assays of a spacecraft by simulating the microbial accretion during periods when assays are not taken; (2) minimize the necessity for a large number of microbiological assays; and (3) predict the microbial loading on a lander immediately prior to sterilization and other non-lander equipment prior to launch. It is shown that these purposes not only were achieved but also that the prediction results compare favorably to the estimates derived from the direct assays. The computer program can be applied not only as a prediction instrument but also as a management and control tool. The basic logic of the model is shown to have possible applicability to other sequential flow processes, such as food processing.
Development of a stochastic model of the probability distribution for the random variable representing the number of microorganisms on a surface as a function of time. The first basic principle associated with bioburden estimation is that viable particles are removed from surfaces. The second notion important to the analysis is that microorganisms in environments and on surfaces occur in clumps. The last basic principle relating to bioburden modeling is that viable particles are deposited on a surface. The bioburden on a spacecraft is determined by the amount and kind of control exercised on the spacecraft assembly location, the shedding characteristics of the individuals in the vicinity of the spacecraft, its orientation, the geographical location in which the assembly takes place, and the steps in the assembly procedure. The model presented has many of the features which are desirable for its use in the spacecraft sterilization programs currently being planned by NASA.
A model of the petroleum exploration process that tests empirically the hypothesis that at an early stage in the exploration of a basin, the process behaves like sampling without replacement is proposed along with a model of the spatial distribution of petroleum reserviors that conforms to observed facts. In developing the model of discovery, the following topics are discussed: probabilitistic proportionality, likelihood function, and maximum likelihood estimation. In addition, the spatial model is described, which is defined as a stochastic process generating values of a sequence or random variables in a way that simulates the frequency distribution of areal extent, the geographic location, and shape of oil deposits
A method is presented for calculating the statistics of the natural frequencies and mode shapes of vibration for a structure acted upon by an external static loading which results in the structure being stressed for eigenvalue analysis. The analytical tools presented apply to the probabilistic eigenvalue problem, and it is apparent that structural parameter uncertainty will significantly influence all aspects of the structure's response. The treatment of a sample problem serves the purpose of furthering understanding for the importance of considering structural parameters as random variables.
The joint distribution (as n tends to infinity) of the maxima of a sample of n independent observations of a bivariate random variable (X,Y) is studied. A method is developed for deriving the asymptotic distribution of the maxima, assuming that X and Y possess asymptotic extreme-value distributions and that the probability element dF(x,y) can be expanded in a canonical series. Applied both to the bivariate normal distribution and to the bivariate gamma and compound correlated bivariate Poisson distributions, the method shows that maxima from all these distributions are asymptotically uncorrelated.
Common to all nondestructive type testing of hardware that provides the necessary confidence in the design is the question regarding the proper magnitude of the test level. The objective of this investigation was to study the possibility of establishing cost optimized test levels of a rather general nature. This investigation was based on studying the influence of test level on a cost of error cost model that reflects the adverse effects of 'undertesting' and 'overtesting'. For the assumed conditions of normal distribution for the random variables and a protoflight spacecraft case, the results indicated that the limits for an optimum test factor can range between slightly less than 1.0 to slightly more than 1.5.
A terminal guidance and navigation scheme developed in earlier work was modified and evaluated for a solar electric propulsion rendezvous mission to comet Encke. The scheme is intended for autonomous, on-board use. The guidance algorithm is based on optimal control theory and minimizes the time integrated square of thrust acceleration. The navigation algorithm employs a modified Kalman filter set in measurement variables. Random sequences were generated to simulate measurement errors, and the evaluation was conducted with detailed numerical computations which include actual motions of spacecraft and comet. The evaluations showed that the scheme attains rendezvous and maintains station after rendezvous within less than 10 km for estimated best measurements and within less than 100 km for estimated worst measurements. The measurements required are angles, range, and range rate. Angles and range appear to be absolutely necessary; range rate is not as strong a measurement type, and further modifications of the filter will allow a scheme that does not require the rate measurements.
The development of reliability-based optimum inspection and maintenance schedules for engines needs an understanding of the fatigue behavior of the engines. Critical areas of the engine structure prone to fatigue damage are usually identified beforehand or after the fleet has been put into operation. In these areas, fatigue cracks initiate after several flight hours, and these cracks grow in length until failure takes place when these cracks attain the critical lengths. Crack initiation time and its growth rate are considered to be random variables. Usually, the inspection (fatigue) or test data from similar engines are used as prior distributions. The existing state-of-the-art is to ignore the different lengths of cracks obserbed at various inspections and to consider only the fact that a crack existed (or did not exist) at the time of inspection. In this paper, a procedure has been developed to obtain the probability of finding a crack of a given size at a certain time if the probability distributions for crack initiation and rates of growth are known. Application of the developed stochastic models to devise optimum procedures for inspection and maintenance are also discussed.