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A directed graph is simple if each ordered pair of vertices is the head and tail of at most one edge; one loop may be present at each vertex. For each n ≥ 1, prove or disprove the following statement. Every simple directed graph with n vertices has two vertices with the same outdegree or two vertices with the same indegree.
(a) How many words of length 2 are there over the alphabet A = { 0; 1; 2; 3}?
(b) Construct the de Bruijn digraph D4,2 and use this digraph to nd an appropriate de Bruijn sequence.
(c) For b = 4 and n = 2, how many de Bruijn sequences can be formed?
Suppose [S, T] and [X, Y ] are two minimal cuts in a network N. Prove that both [S [X U T ∩ Y ] and [S ∩ X, T U Y ] are also minimal cuts in N.
A unified approach to all problems of prediction, estimation, and hypothesis testing. It is based on concept of the decision function, which tells the performer of experiment how t
Johnson-Neyman technique: The technique which can be used in the situations where analysis of the covariance is not valid because of the heterogeneity of slopes. With this method
Kalman filter : A recursive procedure which gives an estimate of the signal when only the 'noisy signal' can be observed. The estimate is efficiently constructed by putting the exp
Random allocation is a technique for creating the treatment and control groups particularly in accordance of the clinical trial. Subjects receive the active treatment or the place
Graduation is the term is employed most often in the application of the actuarial statistics to denote procedures by which the set or group of observed probabilities is adjusted t
Intercropping experiments are the experiments including growing two or more crops at same time on the same patch of land. The crops are not required to be planted nor harvested at
Bivariate boxplot : A bivariate analogue of boxplot in which the inner area contains 50%of the data, and a 'fence' helps to identify the potential outliers. Robust methods or techn
Geo statistics: The body of methods useful for understanding and modelling spatial variability in a course of interest. Central to these techniques is the idea that measurements t
Particlefilters is a simulation method for tracking moving target distributions and for reducing computational burden of the dynamic Bayesian analysis. The method uses a Markov ch
The probability distribution of the various observations is required to obtain the run of two successes in the series of Bernoulli trials with the probability of success equal to a
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